# Relativistic Short Range Phenomena and Space-Time Aspects of Pulse Measurements

Authors: Ezzat G. Bakhoum; Cristian Toma
Journal: Mathematical Problems in Engineering
 (2008)

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## Abstract

Particle physics is increasingly being linked to engineering applications via electron microscopy, nuclear instrumentation, and numerous other applications. It is well known that relativistic particle equations notoriously fail over very short space-time intervals. This paper introduces new versions of Dirac_s equation and of the Klein-Gordon equation that are suitable for
short-range phenomena. Another objective of the paper is to demonstrate
that pulse measurement methods that are based on the wave nature of matter
do not necessarily correlate with physical definitions that are based on
the corpuscular nature of particles.


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## Body

## 1. Introduction

The theory of
special relativity plays a great role in particle physics. Now, particle
physics is increasingly being linked to engineering applications, via electron
microscopy, superconductivity, nuclear instrumentation, to name a few
applications. Since relativistic formulae are at the heart of all such
applications, then it becomes important to find ways to perform numerical
computations related to localized (short-range) relativistic phenomena. For
instance, it is well known that the relativistic version of Schrödinger's
equation, namely, the Dirac equation, cannot normally be solved over a short
interval because it always predicts that the velocity of the electron is equal
toc,
or the speed of light. In applications such as electron microscopy, it becomes
therefore usually necessary to abandon the relativistic formulae and rely
solely on the classical theory of electromagnetism. It is therefore clear that
there is a need at the present time to formulate the Dirac and other
relativistic equations in a manner that allows the computation of short-range phenomena.
This is the first objective of this paper.The second objective of this paper is to show that
space-time measurements on closed-loop trajectories in special relativity and
noncommutative properties of operators in quantum physics require a more rigorous
definition of the method of measurement of interaction phenomena. The use of
the least action principle, for instance, implies some logic definitions for
measuring methods that are based on waves and for measuring methods that are
based on the corpuscular aspects of matter. When measurement is applied to
pulses, those logic definitions include considerations about a possible memory
of previous measurements (space-time operators). Accordingly, a distinction
exists between the set of existing space-time intervals and the set of measured
space-time intervals (established using wave measurement methods and defined
within limited space-time intervals).


## 2. Relativistic Short-Range Electron Equation

In this section, we will develop a version of Dirac's
equation that is suitable for pulsed, short-range electron beams. We will rely
on the recently introduced mass-energy equivalence relationH=mv2 [1] (where H is the total energy of the electron and v is its velocity), which has proved to be
effective in explaining short-range phenomena. First, a new Hamiltonian will be
obtained. It will be then verified that the new Hamiltonian directly leads to
the result that the velocity of the electron must be equal to ±v,
which is of course a result that is in sharp contrast with Dirac's result and
which does agree with experimental observation. We will also verify that the
spin magnetic moment term obtained by Dirac remains unchanged in the present
formulation.


### 2.1. The Wave Equation

We will begin by describing briefly Dirac's approach
for obtaining the relativistic wave equation and then proceed to derive the
modified equation and hence the modified Hamiltonian. Dirac considered the mass
of the particle as represented by its relativistic expressionm=m0/1−v2/c2.
If we square that expression and rearrange the terms, we get(2.1)m2c2=m2v2+m02c2.Multiplying byc2,
we get(2.2)m2c4=m2v2c2+m02c4.But sincemc2 is the total energy according to Einstein,
then we have(2.3)H2=p2c2+m02c4.Hence,(2.4)H=cp2+m02c2.Since the termp2 can be written as ∑rpr2,
where pr is a one-dimensional momentum component and r=1,2,3,
we finally have(2.5)H=c∑rpr2+m02c2.This was Dirac's total energy equation and was
subsequently used to obtain the relativistic wave equation. To obtain the
modified wave equation, we now proceed to multiply (2.1) by v2,
getting(2.6)m2c2v2=m2v4+m02c2v2.UsingH=mv2 as the total energy of the particle, we have,
from the above expression,(2.7)H2=p2c2−m02c2v2.Now, sincev2=∑rvr2,
where vr is a one-dimensional velocity component, (2.7)
can equivalently be written as(2.8)H=c∑rpr2−m02∑rvr2.Equation (2.8) can be further simplified by noting that vr=pr/m.
We finally have(2.9)H=c(1−m02m2)∑rpr2.Following Dirac's approach, if we letp→0 be a vector defined as p→0=H→/c,
where H→ may be Hamiltonian of the form H→=(H,0,0),
we will seek a wave equation that is linear in p→0.
We will take an equation of the most simple, basic form(2.10)(p→0−∑rp→r[αr])ψ=0.This form can be sufficient without any additional
terms if we do not impose any restrictions on the matrices[αr].
Dirac found that such matrices must be noncommuting, but it is obvious here
that such matrices must also contain mass terms. Multiplying (2.10) by the vector (p→0+∑rp→r[αr]),
we get(2.11)p02−(∑rp→r[αr])2=0.Comparing this last expression with (2.9), we conclude
that(2.12)(1−m02m2)∑rpr2=(∑rp→r[αr])2=∑rp→r[αr]2p→rT+∑j∑kp→j[αj][αk]p→kT, where j,k=1,2,3,
and j≠k.
Accordingly, the matrices [αr] must satisfy(2.13)[αr]=±1−m02m2[βr], where [βr] are matrices that must satisfy the following
two conditions:(2.14)[βr]2=I,[βj][βk]+[βk][βj]=0.Examples of such matrices were suggested by
Dirac [2]. They might
take the following forms among others:(2.15)(0001001001001000),(000−i00i00−i00i000).(Note that Dirac used4×4 matrices to account for time as the fourth
dimension. It was independently confirmed later that the minimum number of
dimensions that will satisfy Dirac's theory is in fact four.)Using the relativistic expression form,
the matrices [αr] can now be written as(2.16)[αr]=±1−(1−v2c2)[βr]=±vc[βr].Therefore, from (2.12) and (2.16), the vector Hamiltonian
can be written as(2.17)H→=cp→0=c∑rp→r[αr]±v∑rp→r[βr].To check the modified theory, it can be now easily
verified that the velocity componentx˙1 will be given by(2.18)x˙1=[x1,H→]=±v.Unlike Dirac's result, this result is of course in
agreement with experimental observation. It is important to note here that,
mathematically,x˙1 is the “expected” value of the velocity. From
(2.17), we can also see that the negative energy states are still preserved here.


### 2.2. Motion of a Charged Particle in a Magnetic Field

We now consider the motion of a charged particle in a
magnetic field to obtain a formulation for the spin magnetic moment term that
must appear in the final Hamiltonian (we assume the absence of an electrostatic
field here). In the presence of a magnetic field, the change in the particle
momentumΔp that occurs as a result of the interaction
with the field is given by [3](2.19)Δp=ecA, where e is the particle charge and A is the magnitude of the vector magnetic
potential. Adding that term to the momentum in (2.17) gives the
Hamiltonian(2.20)H→=±v∑r(p→r+ecA→r)[βr].By squaring (2.20), we get(2.21)H2v2=∑r[(p→r+ecA→r)[βr]]2+∑j∑k(p→j+ecA→j)[βj][βk](p→k+ecA→k)T.It is fairly easy to verify that the second term on
the r.h.s. of the above expression must vanish since thep→r vectors commute and since the [βr] matrices satisfy condition (2.14). In Dirac's
treatment of the subject, he was able to show that the following equation
holds:(2.22)[(p→r+ecA→r)[βr]]2(p→r+ecA→r)2+ℏec∥M→[βr]∥, where M→=curl⁢A→ is the magnetic field intensity vector. Equation (2.21)
therefore becomes(2.23)H2=v2∑r(p→r+ecA→r)2+v2ℏec∑r∥M→[βr]∥.(Note that (2.23) is a scalar equation.) If we now let H=mv2 and divide both sides of the equation by 2mv2,
we get(2.24)12mv2=12m∑r(p→r+ecA→r)2+ℏe2mc∑r∥M→[βr]∥.If the particle is an electron, thene is a negative quantity and the above equation
becomes(2.25)12m∑r(p→r+ecA→r)2=12mv2+ℏ|e|2mc∑r∥M→[βr]∥.Without the presence of the magnetic field, the l.h.s.
of (2.25) is reduced to(2.26)12m∑rpr2=p22m.This is the same as1/2mv2.
We can therefore conclude that the second term on the r.h.s. of (2.25) is the
term that represents the interaction of the field with the electron magnetic
moment. Hence the quantity ℏ|e|/2mc is the spin magnetic moment coefficient. In
general, we can withdraw here the following two important conclusions: (1) the
modified theory fully yielded the classical expression of kinetic energy with
the addition of the spin interaction term; and (2) the spin interaction term
obtained here is the same as the one obtained by Dirac [2] (which is one Bohr
magneton). The second conclusion is a confirmation that this part of Dirac's
theory was correct. The first conclusion, however, shows a fact that was not
apparent from Dirac's theory. Specifically, when M→=0 (i.e., when the particle is away from the
magnetic field lines), (2.25) becomes(2.27)12m∑r(p→r+ecA→r)2=12mv2.This is a direct confirmation of the Aharonov-Bohm
effect [4, 5]. Clearly, (2.27) shows that
the components pr of the momentum will be altered while the
kinetic energy remains constant.


## 2.1. The Wave Equation

We will begin by describing briefly Dirac's approach
for obtaining the relativistic wave equation and then proceed to derive the
modified equation and hence the modified Hamiltonian. Dirac considered the mass
of the particle as represented by its relativistic expressionm=m0/1−v2/c2.
If we square that expression and rearrange the terms, we get(2.1)m2c2=m2v2+m02c2.Multiplying byc2,
we get(2.2)m2c4=m2v2c2+m02c4.But sincemc2 is the total energy according to Einstein,
then we have(2.3)H2=p2c2+m02c4.Hence,(2.4)H=cp2+m02c2.Since the termp2 can be written as ∑rpr2,
where pr is a one-dimensional momentum component and r=1,2,3,
we finally have(2.5)H=c∑rpr2+m02c2.This was Dirac's total energy equation and was
subsequently used to obtain the relativistic wave equation. To obtain the
modified wave equation, we now proceed to multiply (2.1) by v2,
getting(2.6)m2c2v2=m2v4+m02c2v2.UsingH=mv2 as the total energy of the particle, we have,
from the above expression,(2.7)H2=p2c2−m02c2v2.Now, sincev2=∑rvr2,
where vr is a one-dimensional velocity component, (2.7)
can equivalently be written as(2.8)H=c∑rpr2−m02∑rvr2.Equation (2.8) can be further simplified by noting that vr=pr/m.
We finally have(2.9)H=c(1−m02m2)∑rpr2.Following Dirac's approach, if we letp→0 be a vector defined as p→0=H→/c,
where H→ may be Hamiltonian of the form H→=(H,0,0),
we will seek a wave equation that is linear in p→0.
We will take an equation of the most simple, basic form(2.10)(p→0−∑rp→r[αr])ψ=0.This form can be sufficient without any additional
terms if we do not impose any restrictions on the matrices[αr].
Dirac found that such matrices must be noncommuting, but it is obvious here
that such matrices must also contain mass terms. Multiplying (2.10) by the vector (p→0+∑rp→r[αr]),
we get(2.11)p02−(∑rp→r[αr])2=0.Comparing this last expression with (2.9), we conclude
that(2.12)(1−m02m2)∑rpr2=(∑rp→r[αr])2=∑rp→r[αr]2p→rT+∑j∑kp→j[αj][αk]p→kT, where j,k=1,2,3,
and j≠k.
Accordingly, the matrices [αr] must satisfy(2.13)[αr]=±1−m02m2[βr], where [βr] are matrices that must satisfy the following
two conditions:(2.14)[βr]2=I,[βj][βk]+[βk][βj]=0.Examples of such matrices were suggested by
Dirac [2]. They might
take the following forms among others:(2.15)(0001001001001000),(000−i00i00−i00i000).(Note that Dirac used4×4 matrices to account for time as the fourth
dimension. It was independently confirmed later that the minimum number of
dimensions that will satisfy Dirac's theory is in fact four.)Using the relativistic expression form,
the matrices [αr] can now be written as(2.16)[αr]=±1−(1−v2c2)[βr]=±vc[βr].Therefore, from (2.12) and (2.16), the vector Hamiltonian
can be written as(2.17)H→=cp→0=c∑rp→r[αr]±v∑rp→r[βr].To check the modified theory, it can be now easily
verified that the velocity componentx˙1 will be given by(2.18)x˙1=[x1,H→]=±v.Unlike Dirac's result, this result is of course in
agreement with experimental observation. It is important to note here that,
mathematically,x˙1 is the “expected” value of the velocity. From
(2.17), we can also see that the negative energy states are still preserved here.


## 2.2. Motion of a Charged Particle in a Magnetic Field

We now consider the motion of a charged particle in a
magnetic field to obtain a formulation for the spin magnetic moment term that
must appear in the final Hamiltonian (we assume the absence of an electrostatic
field here). In the presence of a magnetic field, the change in the particle
momentumΔp that occurs as a result of the interaction
with the field is given by [3](2.19)Δp=ecA, where e is the particle charge and A is the magnitude of the vector magnetic
potential. Adding that term to the momentum in (2.17) gives the
Hamiltonian(2.20)H→=±v∑r(p→r+ecA→r)[βr].By squaring (2.20), we get(2.21)H2v2=∑r[(p→r+ecA→r)[βr]]2+∑j∑k(p→j+ecA→j)[βj][βk](p→k+ecA→k)T.It is fairly easy to verify that the second term on
the r.h.s. of the above expression must vanish since thep→r vectors commute and since the [βr] matrices satisfy condition (2.14). In Dirac's
treatment of the subject, he was able to show that the following equation
holds:(2.22)[(p→r+ecA→r)[βr]]2(p→r+ecA→r)2+ℏec∥M→[βr]∥, where M→=curl⁢A→ is the magnetic field intensity vector. Equation (2.21)
therefore becomes(2.23)H2=v2∑r(p→r+ecA→r)2+v2ℏec∑r∥M→[βr]∥.(Note that (2.23) is a scalar equation.) If we now let H=mv2 and divide both sides of the equation by 2mv2,
we get(2.24)12mv2=12m∑r(p→r+ecA→r)2+ℏe2mc∑r∥M→[βr]∥.If the particle is an electron, thene is a negative quantity and the above equation
becomes(2.25)12m∑r(p→r+ecA→r)2=12mv2+ℏ|e|2mc∑r∥M→[βr]∥.Without the presence of the magnetic field, the l.h.s.
of (2.25) is reduced to(2.26)12m∑rpr2=p22m.This is the same as1/2mv2.
We can therefore conclude that the second term on the r.h.s. of (2.25) is the
term that represents the interaction of the field with the electron magnetic
moment. Hence the quantity ℏ|e|/2mc is the spin magnetic moment coefficient. In
general, we can withdraw here the following two important conclusions: (1) the
modified theory fully yielded the classical expression of kinetic energy with
the addition of the spin interaction term; and (2) the spin interaction term
obtained here is the same as the one obtained by Dirac [2] (which is one Bohr
magneton). The second conclusion is a confirmation that this part of Dirac's
theory was correct. The first conclusion, however, shows a fact that was not
apparent from Dirac's theory. Specifically, when M→=0 (i.e., when the particle is away from the
magnetic field lines), (2.25) becomes(2.27)12m∑r(p→r+ecA→r)2=12mv2.This is a direct confirmation of the Aharonov-Bohm
effect [4, 5]. Clearly, (2.27) shows that
the components pr of the momentum will be altered while the
kinetic energy remains constant.


## 3. Phase and Group Velocities of Short-Range Electrons

The concepts of the phase velocity and the group
velocity are very important concepts that come into play when short-range
phenomena are considered. For instance, de Broglie's work predicts that the
phase velocity of a matter wave is given by the expressionc2/v,
which is a very unrealistic assumption for short-range, slow electrons. We
will attempt in this section to give a better explanation for that problem.First of all, we must realize that there exists a
number of phase velocities, not a single phase velocity. Now, it is well known
mathematically that each phase velocityvp=ωi/ki and that the group velocity vg=dω/dk (where ω is the angular frequency and k=2π/λ is the propagation constant). As was pointed out
in [1], the two
fundamental relationships of wave mechanics, λ=h/p and H=hν,
together make a statement about
the total energy of a particle; that is, H=(pλ)ν=pu,
where u is some velocity. The question here is what is u exactly? Is it a phase velocity or a group
velocity? Apart from the fact that H=pu is a total energy equation, we must also note,
since H=ℏω and p=ℏk,
that the equation leads to the relationship ω=ku.
Hence we must conclude that(3.1)dωdk=ωk=u.This means that the group and the phase velocities arethe same. This is the conclusion
that we must hold as true for short-range phenomena. Let us now attempt to
understand the origin of the problem. De Broglie's original derivation of the
important relationship λ=h/p can be found in a number of standard
references (see, e.g., [6]). Amazingly, as we will conclude, while the formula
was correct, the approach that was used to derive it was not.De Broglie started by assuming a wave function that
describes a stationary particle of the formψ'=exp⁡(iω't').
By using the Lorentz transformation of time t'=γ(t−vx/c2),
then ψ'=exp⁡(iγω'[t−vx/c2]).
Since this equation (in principle) is a traveling wave equation, de Broglie
then concluded that the quantity c2/v must represent the velocity of the wave in the
observer frame. The rest of the derivation that leads to the formula λ=h/p is then straightforward and consists of
letting H=hν=mc2 and substituting the product λν for the quantity c2/v.
As it is well known historically [7, 8], de Broglie later offered the hypothesis that c2/v is only a “phase” velocity and that the
real, or “group” velocity is actually v so that the particle and its associated wave
would not part company. However, as we indicated, the problem with such a
hypothesis is that it directly contradicts the simple conclusion in (3.1).Let us try to understand the problem with the above
approach that led to the indicated contradictions. The Lorentz transformation
of timet'=γ(t−vx/c2),
which includes the coordinate x,
strictly assumes that “x” is only one geometrical point. From the
viewpoint of a stationary observer, a traveling wave, in the observer's frame,
cannot be described by one “x” coordinate. The correct approach for
including a traveling wave within the relativistic transformations is to assume
first that the “x” coordinate is equal to zero (and hence
the time transformation will be t'=γt) and then write a true traveling wave equation in the
observer frame, that is,(3.2)ψ=exp⁡i(kx−ωt).This was indeed the approach that was taken by
Shrödinger and certainly this explains why Shrödinger's equation has been
unquestionably successful. Now, by  noting thatk=2π/λ and ω=2πν,ψ can be written as(3.3)ψ=exp⁡i(2πλx−ωt)=exp⁡iω(2πλx2πν−t)=exp⁡iω(xλν−t).Assume first that the particle is moving with a
velocityv≪c so that the relativistic effects can be
ignored. In this case, ordinary (nonrelativistic) wave mechanics state that λν=v,
or the wave velocity. Now, if the relativistic effect is to be included, then
the wavelength λ becomes λ/γ (length contraction) and the frequency ν becomes γν (frequency shift). The result therefore is
that λν is still equal to v.
We can see, then, that the flaw in the original approach that led to the result λν=c2/v was the incorrect use of the Lorentz
transformation.If we now follow the rest of de Broglie's derivation,
but useH=mv2 instead of mc2,
we have H=mv2=hν,
hence(3.4)p=mv=hνv=hνλν=hλ,which is of course de Broglie's well-known formula. De
Broglie was aware that this relationship can be derived in a number of
different ways, and for that reason he raised it to the level of a postulate. Concerning the approach that
was used in deriving it, however, this is certainly one of the rare cases in
science in which an incorrect derivation procedure still led to the correct
result.


## 4. A Klein-Gordon Equation and a De Broglie Dispersion Relation for Short-Range Electrons

In this section, we present derivations for a modified
Klein-Gordon equation and a modified de Broglie dispersion relation. The
conclusions are: (1) in the case of a massless particle, the dispersion
relation is the same as the original one; and (2) in the case of a massive
particle, we still conclude that the phase and the group velocities are the
same, that is,vg=vp=v.


### 4.1. The Klein-Gordon Equation

The derivation of the Klein-Gordon equation starts
with the usual relativistic expression (see [9])(4.1)H2=p2c2+m02c4.If we now replaceH by mc2 and p by mv,
we have(4.2)m2c4=m2v2c2+m02c4.If we multiply this expression byv2/c2,
we get(4.3)m2v2c2=m2v4+m02v2c2.If we now letH=mv2, we finally have(4.4)H2=p2c2−m02v2c2.This is a modified energy-momentum relationship and
was in fact derived previously in [1]. Notice that the quantity m02v2=p2−H2/c2.
It is therefore a correct representation of the momentum vector pμ.To obtain the modified Klein-Gordon equation, we start
with the well-known relationship(4.5)∇2ψ=−k2ψ=−p2ℏ2ψ.By substituting from (4.4) into (4.5) we
have(4.6)−ℏ2∇2ψ=(H2c2+m02v2)ψ.From Shrödinger's equation we have(4.7)∂2ψ∂t2=−H2ℏ2ψ.By substituting from (4.6) into (4.7) we finally
get(4.8)1c2∂2ψ∂t2−∇2ψ=(m0vℏ)2ψ.This is the modified Klein-Gordon equation.


### 4.2. De Broglie's Dispersion Relation

In view of (4.7) and (4.5), the modified
Klein-Gordon equation can be written as(4.9)−1c2(ω2ℏ2ℏ2)ψ=−k2ψ+(m0vℏ)2ψ,⁢or,⁢ω2ℏ2ψ=c2ℏ2k2ψ−m02c2v2ψ.Hence, the modified de Broglie wave dispersion
relation is(4.10)ℏ2ω2=c2ℏ2k2−m02c2v2.Form0=0,
we can see that the relation becomes ℏ2ω2=c2ℏ2k2,
which is of course the same as in the usual theory.To obtain the group velocity,vg=dω/dk,
we differentiate the dispersion relation with respect to k,
getting (note that only the magnitudes of the vectors p and k will be represented)(4.11)ℏ2ωdωdk=c2ℏ2k−m02c2vdvdk.Sincep=mv=ℏk and hence m(dv/dk)=ℏ,
the above equation becomes(4.12)ℏ2ωdωdk=c2ℏ2k−m02c2ℏ2m2k or(4.13)ωdωdk=c2k−m02m2c2k=c2k(1−(1−v2c2))=kv2.Hence,(4.14)dωdk=(kω)v2.But sincedω/dk=vg=v,
we then conclude that ω/k=vp=v.
The group and the phase velocities are therefore the same.


## 4.1. The Klein-Gordon Equation

The derivation of the Klein-Gordon equation starts
with the usual relativistic expression (see [9])(4.1)H2=p2c2+m02c4.If we now replaceH by mc2 and p by mv,
we have(4.2)m2c4=m2v2c2+m02c4.If we multiply this expression byv2/c2,
we get(4.3)m2v2c2=m2v4+m02v2c2.If we now letH=mv2, we finally have(4.4)H2=p2c2−m02v2c2.This is a modified energy-momentum relationship and
was in fact derived previously in [1]. Notice that the quantity m02v2=p2−H2/c2.
It is therefore a correct representation of the momentum vector pμ.To obtain the modified Klein-Gordon equation, we start
with the well-known relationship(4.5)∇2ψ=−k2ψ=−p2ℏ2ψ.By substituting from (4.4) into (4.5) we
have(4.6)−ℏ2∇2ψ=(H2c2+m02v2)ψ.From Shrödinger's equation we have(4.7)∂2ψ∂t2=−H2ℏ2ψ.By substituting from (4.6) into (4.7) we finally
get(4.8)1c2∂2ψ∂t2−∇2ψ=(m0vℏ)2ψ.This is the modified Klein-Gordon equation.


## 4.2. De Broglie's Dispersion Relation

In view of (4.7) and (4.5), the modified
Klein-Gordon equation can be written as(4.9)−1c2(ω2ℏ2ℏ2)ψ=−k2ψ+(m0vℏ)2ψ,⁢or,⁢ω2ℏ2ψ=c2ℏ2k2ψ−m02c2v2ψ.Hence, the modified de Broglie wave dispersion
relation is(4.10)ℏ2ω2=c2ℏ2k2−m02c2v2.Form0=0,
we can see that the relation becomes ℏ2ω2=c2ℏ2k2,
which is of course the same as in the usual theory.To obtain the group velocity,vg=dω/dk,
we differentiate the dispersion relation with respect to k,
getting (note that only the magnitudes of the vectors p and k will be represented)(4.11)ℏ2ωdωdk=c2ℏ2k−m02c2vdvdk.Sincep=mv=ℏk and hence m(dv/dk)=ℏ,
the above equation becomes(4.12)ℏ2ωdωdk=c2ℏ2k−m02c2ℏ2m2k or(4.13)ωdωdk=c2k−m02m2c2k=c2k(1−(1−v2c2))=kv2.Hence,(4.14)dωdk=(kω)v2.But sincedω/dk=vg=v,
we then conclude that ω/k=vp=v.
The group and the phase velocities are therefore the same.


## 5. Logical Aspects Connected with Space-Time Measurements

After presenting basic aspects in physics from the
relativistic point of view, we will present some logical aspects for basic
principles in physics (the principle of constant light in vacuum in any
reference system and the uncertainty principle in quantum theory). We will show
that these principles make use in an implicit manner of terms which are
defining also the conclusion. For example, the idea of constant light speed
implies the use of a measuring method based on a clocks' synchronization
performed using a supposed antecedent light signal transmitted and reflected
toward the observer. In a similar manner, the uncertainty principle implies the
existence of a measuring method for position or time correlated with a
subsequent measurement for momentum or energy (measurements which also make use
of position and time). Yet a logic definition of a physics principle cannot be
based on the use of the same terms in both sides of it; like in the case of an
algebraic calculus, the quantity to be determined must be finally placed in the
opposite side of an equality, as related to the already known quantities joined
in a mathematical operation. More precisely, we cannot define in a rigorous
manner a certain term using the same term in the corresponding definition.


### 5.1. Logical Aspects of Light Speed Constance Principle

The constant light speed principle (in vacuum) can be
considered under the following form. It exists a quantitylight speed in vacuum noted as c, which is constant for any observer inside an inertial reference
system.We can notice at first step that in an implicit manner
the previous definition requires the existence of a measuring method for light
speed in vacuum; any method for measuring a speed requires the use of time
measurements (whilev=Δr/Δt). For our case (special relativity theory),
the correspondence of time moments in different reference systems is based on a
previous synchronization procedure implying an emission of light from an
observer to another and a reflection of this light signal from the other observer
to the first one. The reflection moment (considered as synchronization moment ≡ zero moment) is considered by the first
observer to take place at the middle of the time interval between the initial
emission of light and the return of it. The whole chain implies that the use of
a wave light LW appears in the definition of the light speed
constance principle (in vacuum) under an explicit form (the notion of light
speed), and it appears also under an implicit form (a previous synchronization
based on light signals is required). From the formal logic point of view, this
represents a contradiction [10]. A first attempt to solve it would be in taking into account
the fact that the light speed measurement and the systems synchronization
correspond to different time moments (the light wave considered for systems
synchronization corresponds to the zero moment of time, while the light wave
whose speed is considered in light speed constance principle corresponds to a
subsequent moment of time).However, the use of such a set of different light
waves (a light wave whose speed has to be measured and a previous pair of
emitted-reflected light wave necessary for the synchronization procedure) implies
the use of an extended time interval required by a light speed measurement as(5.1)Tm=[t0,tm],where Tm is the time interval required by a light speed
measurement at tm time moment. But at next step we can notice
that a speed corresponds to an almost instant moment of time, being defined
as(5.2)v=lim⁡t→tmΔrΔt=drdt.This requires that the time
interval required by a speed measurement must be infinitely small. Thus the
time interval necessary for light speed measurement can be written
as(5.3)Tm=[tm−Δt,tm]which implies that the
corresponding length interval LTm is infinitely small(5.4)LTm→0.But this is in contradiction with
the previous consideration Tm=[t0,tm].
The corresponding timelength(5.5)LTm=tm−t0≫0can be much greater than zero.
So the contradiction can be easily proved as(5.6)LTM→0⁢and⁢in⁢the⁢same⁢time⁢LTm≫0.From the intuitive point of
view, this means that a light wave emitted in a certain reference system
interacts in the most general case only on a limited time interval with another
measuring reference system, the use of a previous procedure of
emission-reflection for synchronization being impossible in practice. So the
solution of such a contradiction (determined by implicit aspects of the terms
used in definitions) must be found by taking into consideration other
properties of physics entities involved in definition; see also [11].


### 5.2. Logical Aspects of Uncertainty Principle in Quantum Mechanics

If we study the uncertainty principle in quantum
mechanics, we can notice quite similar aspects. According to this principle, a
measurement performed with a greater accuracy upon space or time coordinates
for a quantum particle must generate a greater error upon a subsequent
measurement for momentum or energy according to(5.7)ΔxΔPx≥h4πor(5.8)ΔtΔE≥h4π.But the existence of a measuring
method for position or time is correlated with a subsequent measurement for
momentum or energy (measurements which also make use of position and time).It can be noticed that a term (a space-time
measurements) is explained using (in an implicit manner) the same term at a
subsequent moment of time. Without being a contradiction (like in the case of
light speed constance principle), it still remains a recurrent definition. In
the same manner presented for special relativity, we can take into
consideration the different moments of time for space-time measurements. Yet
the fact that (in an implicit manner) the principle requires the use of a
measurement performed at a later time moment generates another logical problem.
Can a space-time measurement performed at a certain moment of time be
influenced by previous space-time measurements performed upon the same quantum
particle? When a space-time measurement belongs to the class of space-time
coordinates measurement, and when it belongs to the class of momentum or energy
measurements (performed in an indirect manner using also space-time
measurements)? Under which circumstances a measurement can be considered as an
initial action (in this case its accuracy can be greater) or as a subsequent
action (its accuracy having to be less than a certain value, according to
Heisenberg relation)?. The time always appears in quantum mechanics, while two
physical quantities cannot be measuredexactly at the same moment of time.So a space or time measurement performed at a certain
time moment belongs to the class of subsequent indirect methods for measuring
momentum or energy (having as a consequence a limited accuracy), or to the
class of direct methods for measuring space or time (having a possible greater
accuracy). A rigorous classification according to certain patterns should be
made; see also [12],
taking into consideration similarities in fundamental physics
laws [13].


### 5.3. Different-Scale System Properties Used for Explaining Logical Aspects of Pulse Measurements

This problem suggests also a possible solution: if we
continue our analysis of terms involved in measuring procedures, we can notice
that both basic principles (light speed constance principle and uncertainty
principle) use the term of measuring method. In an implicit manner, the termsreference system (for special relativity theory) and measuring system (for quantum theory) appear. Yet a measuring
system implies the fact that it is not affected by the measuring procedure
(otherwise, the physical quantity having to be measured would possess different
values, depending on this interaction). So a first conclusion appears: the
measuring system must be defined at a much larger scale than the body or the
wave which interacts with it. The different scale system properties must be
taken into consideration from the very beginning so as to put them into
correspondence with(i)
the class of reference systems, which are not
affected by interaction (where wave trains similar to wavelets can appear
[14]);(ii)
the class of transient phenomena which undergo
specific interactions (such transient phenomena can be represented as solitary
waves, while estimations for the space coordinates for the source of the
received wave-train based on space relations are not suitable for this purpose.
As a further consequence, the constance light speed principle appears as a
simple generation of another light wave when a received wave train arrives in
thematerial medium of the observer reference system, and
the uncertainty principle appears as a spread of a wave corresponding to a
quantum particle by the measuring system, according to a kind of Fourier
transformation performed on limited space and time intervals (the aperture and
a certain working period). Thus logical aspects of the definitions of basic
principles in physics (implying measurements of pulse parameters) can be
explained in a rigorous manner.


## 5.1. Logical Aspects of Light Speed Constance Principle

The constant light speed principle (in vacuum) can be
considered under the following form. It exists a quantitylight speed in vacuum noted as c, which is constant for any observer inside an inertial reference
system.We can notice at first step that in an implicit manner
the previous definition requires the existence of a measuring method for light
speed in vacuum; any method for measuring a speed requires the use of time
measurements (whilev=Δr/Δt). For our case (special relativity theory),
the correspondence of time moments in different reference systems is based on a
previous synchronization procedure implying an emission of light from an
observer to another and a reflection of this light signal from the other observer
to the first one. The reflection moment (considered as synchronization moment ≡ zero moment) is considered by the first
observer to take place at the middle of the time interval between the initial
emission of light and the return of it. The whole chain implies that the use of
a wave light LW appears in the definition of the light speed
constance principle (in vacuum) under an explicit form (the notion of light
speed), and it appears also under an implicit form (a previous synchronization
based on light signals is required). From the formal logic point of view, this
represents a contradiction [10]. A first attempt to solve it would be in taking into account
the fact that the light speed measurement and the systems synchronization
correspond to different time moments (the light wave considered for systems
synchronization corresponds to the zero moment of time, while the light wave
whose speed is considered in light speed constance principle corresponds to a
subsequent moment of time).However, the use of such a set of different light
waves (a light wave whose speed has to be measured and a previous pair of
emitted-reflected light wave necessary for the synchronization procedure) implies
the use of an extended time interval required by a light speed measurement as(5.1)Tm=[t0,tm],where Tm is the time interval required by a light speed
measurement at tm time moment. But at next step we can notice
that a speed corresponds to an almost instant moment of time, being defined
as(5.2)v=lim⁡t→tmΔrΔt=drdt.This requires that the time
interval required by a speed measurement must be infinitely small. Thus the
time interval necessary for light speed measurement can be written
as(5.3)Tm=[tm−Δt,tm]which implies that the
corresponding length interval LTm is infinitely small(5.4)LTm→0.But this is in contradiction with
the previous consideration Tm=[t0,tm].
The corresponding timelength(5.5)LTm=tm−t0≫0can be much greater than zero.
So the contradiction can be easily proved as(5.6)LTM→0⁢and⁢in⁢the⁢same⁢time⁢LTm≫0.From the intuitive point of
view, this means that a light wave emitted in a certain reference system
interacts in the most general case only on a limited time interval with another
measuring reference system, the use of a previous procedure of
emission-reflection for synchronization being impossible in practice. So the
solution of such a contradiction (determined by implicit aspects of the terms
used in definitions) must be found by taking into consideration other
properties of physics entities involved in definition; see also [11].


## 5.2. Logical Aspects of Uncertainty Principle in Quantum Mechanics

If we study the uncertainty principle in quantum
mechanics, we can notice quite similar aspects. According to this principle, a
measurement performed with a greater accuracy upon space or time coordinates
for a quantum particle must generate a greater error upon a subsequent
measurement for momentum or energy according to(5.7)ΔxΔPx≥h4πor(5.8)ΔtΔE≥h4π.But the existence of a measuring
method for position or time is correlated with a subsequent measurement for
momentum or energy (measurements which also make use of position and time).It can be noticed that a term (a space-time
measurements) is explained using (in an implicit manner) the same term at a
subsequent moment of time. Without being a contradiction (like in the case of
light speed constance principle), it still remains a recurrent definition. In
the same manner presented for special relativity, we can take into
consideration the different moments of time for space-time measurements. Yet
the fact that (in an implicit manner) the principle requires the use of a
measurement performed at a later time moment generates another logical problem.
Can a space-time measurement performed at a certain moment of time be
influenced by previous space-time measurements performed upon the same quantum
particle? When a space-time measurement belongs to the class of space-time
coordinates measurement, and when it belongs to the class of momentum or energy
measurements (performed in an indirect manner using also space-time
measurements)? Under which circumstances a measurement can be considered as an
initial action (in this case its accuracy can be greater) or as a subsequent
action (its accuracy having to be less than a certain value, according to
Heisenberg relation)?. The time always appears in quantum mechanics, while two
physical quantities cannot be measuredexactly at the same moment of time.So a space or time measurement performed at a certain
time moment belongs to the class of subsequent indirect methods for measuring
momentum or energy (having as a consequence a limited accuracy), or to the
class of direct methods for measuring space or time (having a possible greater
accuracy). A rigorous classification according to certain patterns should be
made; see also [12],
taking into consideration similarities in fundamental physics
laws [13].


## 5.3. Different-Scale System Properties Used for Explaining Logical Aspects of Pulse Measurements

This problem suggests also a possible solution: if we
continue our analysis of terms involved in measuring procedures, we can notice
that both basic principles (light speed constance principle and uncertainty
principle) use the term of measuring method. In an implicit manner, the termsreference system (for special relativity theory) and measuring system (for quantum theory) appear. Yet a measuring
system implies the fact that it is not affected by the measuring procedure
(otherwise, the physical quantity having to be measured would possess different
values, depending on this interaction). So a first conclusion appears: the
measuring system must be defined at a much larger scale than the body or the
wave which interacts with it. The different scale system properties must be
taken into consideration from the very beginning so as to put them into
correspondence with(i)
the class of reference systems, which are not
affected by interaction (where wave trains similar to wavelets can appear
[14]);(ii)
the class of transient phenomena which undergo
specific interactions (such transient phenomena can be represented as solitary
waves, while estimations for the space coordinates for the source of the
received wave-train based on space relations are not suitable for this purpose.
As a further consequence, the constance light speed principle appears as a
simple generation of another light wave when a received wave train arrives in
thematerial medium of the observer reference system, and
the uncertainty principle appears as a spread of a wave corresponding to a
quantum particle by the measuring system, according to a kind of Fourier
transformation performed on limited space and time intervals (the aperture and
a certain working period). Thus logical aspects of the definitions of basic
principles in physics (implying measurements of pulse parameters) can be
explained in a rigorous manner.


## 6. Aspects Connected with Measurements on a Set of Pulses

### 6.1. Measurements on a Set of Pulses Received on Adjoining Space-Time Intervals.
Synchronization Aspects

We will justify the previous considerations by
presenting the case of measurements for sequence of pulses received on
adjoining space-time intervals. As it is known, the
special relativity theory
considers that the Lorentz formulae describe the transformation of the
space-time coordinates corresponding to an event when the inertial reference
system is changed. These formulae are considered to be valid at any moment of
time after a certain synchronization moment (the zero moment) irrespective to
the measuring method used. However, there are some problems connected to the
use of mechanical measurements on closed-loop trajectories. For example, let us
consider that at the zero moment of time, in a medium with a gravitational
field which can be neglected (the use of the Galileean form of the tensorgik being allowed) two observers are beginning a
movement from the same point of space, in opposite directions, on circular
trajectories having a very great radius of curvature. After a certain time
interval, the observers are meeting again in the same point of space. For very
great radii of curvature, the movements on very small time intervals can be
considered as approximative inertial (as in the case of the transverse Doppler
effect, where the time dilation phenomenon was noticed in the earth reference
system which is approximative inertial on small time intervals). The Lorentz
formulae can be applied on a small time interval Δt(1) measured by one of the observers inside his
reference system, and it results (using the Lorentz formula for time) that this
interval corresponds to a time interval(6.1)Δt′(1)=Δt(1)1−v(1)2/c2in the reference system S2 of the other observer, which moves with speed v(1) as related to the reference system S1 on this time interval. So the time dilation
phenomenon appears. If each observer considers the end of this time interval (Δt(1)orΔt′(1)) as a new zero moment (using a
resynchronization procedure), the end of the second time interval Δt(2) (with the new zero moment considered as
origin) will correspond to a time moment(6.2)Δt′(2)=Δt(1)1−v(2)2/c2measured in the other reference
system S2 which moves with speed v(2) as related to system S1 on the time interval Δt′(2) (with the new zero moment considered as
origin). As related to the first zero moment (when the circular movement has
started) the end of the second time interval appears at the time
moment(6.3)t2=Δt(1)+Δt(2).For the observers situated in reference systemS1,
and at the time moment(6.4)t′(2)=Δt′(1)+Δt′(2)Δt(1)1−v(1)2/c2+Δt(2)1−v(2)2/c2for the other observer.Due to the fact that(6.5)Δt′(1)>Δt(1),Δt′(2)>Δt(2),it results that(6.6)t′(2)=Δt′(1)+Δt′(2)>Δt(1)+Δt(2)=t(2)and  thus a global time dilation
for the time interval Δt(1)+Δt(2) appears. The procedure can continue, by
considering the end of each time interval(6.7)Δt(1)+Δt(2)+⋯+Δt(i)as a new zero moment, and so it
results that on all the circular movement period, a time moment(6.8)t(k)=∑i=0kΔt(i)(measured by the observer in
reference system S1) corresponds to a time moment(6.9)t′(k)=∑i=0kΔt′(i)=∑i=0kΔt(i)1−vi2/c2(measured by the observer
situated in reference system S2) which implies(6.10)t′(k)>t(k).By joining together all these
time intervals Δt(i) we obtain the period of the whole circular
movement T.
While the end of this movement is represented by the end of the time interval Δt(N) in the reference system S1,
it results that T can be written under the form(6.11)T=t(N)=∑i=0NΔt(i)(considered in the reference
system S1) and it results also that this time moment
(the end of the circular movement) corresponds to a time moment(6.12)T′=t′(N)=∑i=0NΔt′(i)measured in the reference system S@.
While(6.13)Δt′(i)=Δt(i)1−v(i)2/c2>Δt(i),it results(6.14)T′>T.If the time is measured using the age of two twin
children, it results that the twin in reference systemS2 is older than the other in reference system S1,
(having a less mechanical resistance of  bones) and it can be destroyed by it
after both observers stop their circular movements. However, the same analysis
can be made starting from another set of small time intervals Δnt′(i) considered in the reference system S2 which corresponds to a new set of time
intervals Δnt(i) considered in the reference system S2 (established using the same Lorentz relation)
and finally it would result that the period of the circular movement T′ measured in system S2 corresponds to a period T greater than T′ considered in reference system S1.
If the time is measured using the age of two twin children, it results that the
twin in reference system S1 is older than the other in reference system S2,
(having a less mechanical resistance of bones) and it can be destroyed by it
after both observers stop their circular movements. But this result is in logic
contradiction with the previous conclusion, because a man cannot destroy and in
the same time be destroyed by another man [15].As a first attempt of solving this contradiction, one
can suppose that Lorentz formulae are valid only for electromagnetic phenomena
(as in the case of the transversal Doppler effect) and not in case of
mechanical phenomena. But such a classification is not a rigorous
classification, being not suitable for formal logic. In next section, we will
present a more rigorous classification of phenomena used in space-time
measurements, which can be used forgedanken experiments using artificial
intelligence based on formal logic.


### 6.2. Classification of Space-Time Measurement Methods
Based on Memory of Previous
Measurements

The logical contradiction presented in previous
section appeared due to the fact that an element with internal memory has been
used. The indication of this element has not been affected by the
resynchronization procedure. In modern physics such an element with internal
memory is connected with the corpuscular aspect of matter with a body. On the
contrary, a measuring procedure based on an electromagnetic or optic wave train
is a transient phenomenon. The synchronization of clocks is possible only after
the wave-train arrives at the observer. Excepting a short time interval after
the reception the received wave train does not exist inside the observer
medium, so there is not any space area where a physical quantity which
characterizes the wave to cumulate. That is the reason why a correct solution
of the twins paradox must be based not on the association of electromagnetic
(or optic) phenomena with the Lorentz formulae, but on the association of the
Lorentz formulae with wave phenomena describing the propagation of a wave
inside the observers reference systems. The wave class is more general than the
class of electromagnetic and optic waves (we can mention the wave associated
with particles in quantum mechanics). Besides, in the most general case, the
interaction between two reference systems appears under the form of a field,
not under the form of a material body. Moreover, this aspect implies an
intuitive interpretation for the dependence of the mass of a body inside a
reference system.Using the formal logic, all we have shown can be
presented in a rigorous manner.(a) We define the notion of
“propagation” phenomenon in two inertial reference systems (the system
where the event takes place and the system where a signal generated by the
event is noticed) as a phenomenon having a finite existence inside the
reference system, the number of intervals being finite.(b) We define the notion of corpuscle inside a certain
reference system as a phenomenon which can possess an unlimited evolution in
time and space inside the reference system; it can be also said that the
phenomenon has its own existence, it exists by itself.(c) We define the emission of a wave-trainUe in a reference system and its transformation
in another train when it interacts with the observers mediumDeffinition 6.1.
There exist  an
areaS0e and a time interval T0e in the reference system where the emission
takes place so that (6.15)Fue(S0e,T0e)≠0,Fue(S0e,t)=0fort⁢∉T0e.There
exist  a space area S0r and a time interval T0r in the observer reference system, and a relation Tr so that(6.16)Fur(S0r,T0r)=Tr[Fue(S0e,T0e)],Fur(S0r,T0r)≠0,Fur(S0r,t)=0fort⁢∉T0r.
(d) We define the transformation of a sequence of
received pulses ΣkUek in a sequence ΣkUrk,k=1⋯n after interaction with the
observers reference system, by considering that each pulse (wave-train) is
transformed in an independent manner by the material medium of the observer
reference system, according to its specific Lorentz transformationDeffinition 6.2.
Consider(6.17)Urk=Lk[Ue]k,ΣkUek=ΣkUrk, where Lk represents the Lorentz transformation
performed upon the Uek wave by the system with the interaction moment
of this wave with the material medium of the observer considered as zero moment
of time (synchronization moment) for the Lorentz transformation Lk.
(e) We define the interaction between a sequence of
pulses and the material body of the observer reference system (a corpuscle) as
an interaction function Int between the material medium and each
transformed pulseUrk corresponding to a received pulse Uek,
the mass m of the body measuring the influence of the
received wave-train Uek upon the body.Deffinition 6.3.
Consider(6.18)1m=Int[Urk]=Int[Lk(Ue)k].When Lorentz transformationLk does not generate a pulse Urk (e.g., when the relative speed between the
material body and the wave is equal to c,
the speed of light in vacuum), the mass m is equal to ∞, which means that no interaction due to the
received pulse Uek exists (an idea which connects the notion on
infinite mass with the absence of interaction). So m=∞ for a body inside a reference system S shows that we cannot act upon the material
body using wave pulses emitted in system S;
however, changes in the movement of the body (considered in system S) due to other external forces seem to be
allowed.By interaction with a certain material medium, each
pulse is transformed according to Lorentz formulae, and the modified parameters
of each pulse must replace the previous informations in the memory cells.


## 6.1. Measurements on a Set of Pulses Received on Adjoining Space-Time Intervals.
Synchronization Aspects

We will justify the previous considerations by
presenting the case of measurements for sequence of pulses received on
adjoining space-time intervals. As it is known, the
special relativity theory
considers that the Lorentz formulae describe the transformation of the
space-time coordinates corresponding to an event when the inertial reference
system is changed. These formulae are considered to be valid at any moment of
time after a certain synchronization moment (the zero moment) irrespective to
the measuring method used. However, there are some problems connected to the
use of mechanical measurements on closed-loop trajectories. For example, let us
consider that at the zero moment of time, in a medium with a gravitational
field which can be neglected (the use of the Galileean form of the tensorgik being allowed) two observers are beginning a
movement from the same point of space, in opposite directions, on circular
trajectories having a very great radius of curvature. After a certain time
interval, the observers are meeting again in the same point of space. For very
great radii of curvature, the movements on very small time intervals can be
considered as approximative inertial (as in the case of the transverse Doppler
effect, where the time dilation phenomenon was noticed in the earth reference
system which is approximative inertial on small time intervals). The Lorentz
formulae can be applied on a small time interval Δt(1) measured by one of the observers inside his
reference system, and it results (using the Lorentz formula for time) that this
interval corresponds to a time interval(6.1)Δt′(1)=Δt(1)1−v(1)2/c2in the reference system S2 of the other observer, which moves with speed v(1) as related to the reference system S1 on this time interval. So the time dilation
phenomenon appears. If each observer considers the end of this time interval (Δt(1)orΔt′(1)) as a new zero moment (using a
resynchronization procedure), the end of the second time interval Δt(2) (with the new zero moment considered as
origin) will correspond to a time moment(6.2)Δt′(2)=Δt(1)1−v(2)2/c2measured in the other reference
system S2 which moves with speed v(2) as related to system S1 on the time interval Δt′(2) (with the new zero moment considered as
origin). As related to the first zero moment (when the circular movement has
started) the end of the second time interval appears at the time
moment(6.3)t2=Δt(1)+Δt(2).For the observers situated in reference systemS1,
and at the time moment(6.4)t′(2)=Δt′(1)+Δt′(2)Δt(1)1−v(1)2/c2+Δt(2)1−v(2)2/c2for the other observer.Due to the fact that(6.5)Δt′(1)>Δt(1),Δt′(2)>Δt(2),it results that(6.6)t′(2)=Δt′(1)+Δt′(2)>Δt(1)+Δt(2)=t(2)and  thus a global time dilation
for the time interval Δt(1)+Δt(2) appears. The procedure can continue, by
considering the end of each time interval(6.7)Δt(1)+Δt(2)+⋯+Δt(i)as a new zero moment, and so it
results that on all the circular movement period, a time moment(6.8)t(k)=∑i=0kΔt(i)(measured by the observer in
reference system S1) corresponds to a time moment(6.9)t′(k)=∑i=0kΔt′(i)=∑i=0kΔt(i)1−vi2/c2(measured by the observer
situated in reference system S2) which implies(6.10)t′(k)>t(k).By joining together all these
time intervals Δt(i) we obtain the period of the whole circular
movement T.
While the end of this movement is represented by the end of the time interval Δt(N) in the reference system S1,
it results that T can be written under the form(6.11)T=t(N)=∑i=0NΔt(i)(considered in the reference
system S1) and it results also that this time moment
(the end of the circular movement) corresponds to a time moment(6.12)T′=t′(N)=∑i=0NΔt′(i)measured in the reference system S@.
While(6.13)Δt′(i)=Δt(i)1−v(i)2/c2>Δt(i),it results(6.14)T′>T.If the time is measured using the age of two twin
children, it results that the twin in reference systemS2 is older than the other in reference system S1,
(having a less mechanical resistance of  bones) and it can be destroyed by it
after both observers stop their circular movements. However, the same analysis
can be made starting from another set of small time intervals Δnt′(i) considered in the reference system S2 which corresponds to a new set of time
intervals Δnt(i) considered in the reference system S2 (established using the same Lorentz relation)
and finally it would result that the period of the circular movement T′ measured in system S2 corresponds to a period T greater than T′ considered in reference system S1.
If the time is measured using the age of two twin children, it results that the
twin in reference system S1 is older than the other in reference system S2,
(having a less mechanical resistance of bones) and it can be destroyed by it
after both observers stop their circular movements. But this result is in logic
contradiction with the previous conclusion, because a man cannot destroy and in
the same time be destroyed by another man [15].As a first attempt of solving this contradiction, one
can suppose that Lorentz formulae are valid only for electromagnetic phenomena
(as in the case of the transversal Doppler effect) and not in case of
mechanical phenomena. But such a classification is not a rigorous
classification, being not suitable for formal logic. In next section, we will
present a more rigorous classification of phenomena used in space-time
measurements, which can be used forgedanken experiments using artificial
intelligence based on formal logic.


## 6.2. Classification of Space-Time Measurement Methods
Based on Memory of Previous
Measurements

The logical contradiction presented in previous
section appeared due to the fact that an element with internal memory has been
used. The indication of this element has not been affected by the
resynchronization procedure. In modern physics such an element with internal
memory is connected with the corpuscular aspect of matter with a body. On the
contrary, a measuring procedure based on an electromagnetic or optic wave train
is a transient phenomenon. The synchronization of clocks is possible only after
the wave-train arrives at the observer. Excepting a short time interval after
the reception the received wave train does not exist inside the observer
medium, so there is not any space area where a physical quantity which
characterizes the wave to cumulate. That is the reason why a correct solution
of the twins paradox must be based not on the association of electromagnetic
(or optic) phenomena with the Lorentz formulae, but on the association of the
Lorentz formulae with wave phenomena describing the propagation of a wave
inside the observers reference systems. The wave class is more general than the
class of electromagnetic and optic waves (we can mention the wave associated
with particles in quantum mechanics). Besides, in the most general case, the
interaction between two reference systems appears under the form of a field,
not under the form of a material body. Moreover, this aspect implies an
intuitive interpretation for the dependence of the mass of a body inside a
reference system.Using the formal logic, all we have shown can be
presented in a rigorous manner.(a) We define the notion of
“propagation” phenomenon in two inertial reference systems (the system
where the event takes place and the system where a signal generated by the
event is noticed) as a phenomenon having a finite existence inside the
reference system, the number of intervals being finite.(b) We define the notion of corpuscle inside a certain
reference system as a phenomenon which can possess an unlimited evolution in
time and space inside the reference system; it can be also said that the
phenomenon has its own existence, it exists by itself.(c) We define the emission of a wave-trainUe in a reference system and its transformation
in another train when it interacts with the observers mediumDeffinition 6.1.
There exist  an
areaS0e and a time interval T0e in the reference system where the emission
takes place so that (6.15)Fue(S0e,T0e)≠0,Fue(S0e,t)=0fort⁢∉T0e.There
exist  a space area S0r and a time interval T0r in the observer reference system, and a relation Tr so that(6.16)Fur(S0r,T0r)=Tr[Fue(S0e,T0e)],Fur(S0r,T0r)≠0,Fur(S0r,t)=0fort⁢∉T0r.
(d) We define the transformation of a sequence of
received pulses ΣkUek in a sequence ΣkUrk,k=1⋯n after interaction with the
observers reference system, by considering that each pulse (wave-train) is
transformed in an independent manner by the material medium of the observer
reference system, according to its specific Lorentz transformationDeffinition 6.2.
Consider(6.17)Urk=Lk[Ue]k,ΣkUek=ΣkUrk, where Lk represents the Lorentz transformation
performed upon the Uek wave by the system with the interaction moment
of this wave with the material medium of the observer considered as zero moment
of time (synchronization moment) for the Lorentz transformation Lk.
(e) We define the interaction between a sequence of
pulses and the material body of the observer reference system (a corpuscle) as
an interaction function Int between the material medium and each
transformed pulseUrk corresponding to a received pulse Uek,
the mass m of the body measuring the influence of the
received wave-train Uek upon the body.Deffinition 6.3.
Consider(6.18)1m=Int[Urk]=Int[Lk(Ue)k].When Lorentz transformationLk does not generate a pulse Urk (e.g., when the relative speed between the
material body and the wave is equal to c,
the speed of light in vacuum), the mass m is equal to ∞, which means that no interaction due to the
received pulse Uek exists (an idea which connects the notion on
infinite mass with the absence of interaction). So m=∞ for a body inside a reference system S shows that we cannot act upon the material
body using wave pulses emitted in system S;
however, changes in the movement of the body (considered in system S) due to other external forces seem to be
allowed.By interaction with a certain material medium, each
pulse is transformed according to Lorentz formulae, and the modified parameters
of each pulse must replace the previous informations in the memory cells.


## 7. Associating a Certain Wave Train to Lorentz Transformation

### 7.1. The Necessity for Associating a Wave Function to the Lorentz Transformation

The Lorentz transformation is usually represented as a
matrixL which acts upon a quadridimensional column
vector r having the components r1=x,⁢r2=y,⁢r3=z,⁢r4=ict,
resulting in  another quadridimensional vector r*
having the components r1′=x′,⁢r2′=y′,⁢r3′=z′,⁢r4′=ict′,
where x,⁢y,⁢z,⁢t are the space-time coordinates corresponding
to a certain event in an inertial reference system S,
and x′,⁢y′,⁢z′,⁢t′ are the space-time coordinates corresponding
to the same event measured in an inertial reference system S′ which moves with velocity v  (a vector) against the system S. This
means(7.1)r′=L(v)r.All time moments are considered after a
synchronization moment (when the clock indications in the reference systems are
set to zero). The velocityv defines the matrix L,
and the result is considered not to depend on the measuring method used. But
let us consider that the velocity v has two components vx and vy oriented along the Ox axis (for vx) and along the Oy axis (for vy) and let us consider also that the event
taking place in the reference system S is first observed in a reference system S1 which moves with velocity vx as against the system S
:
a set of space-time coordinates (x1,y1,z1,t1) will be established for the event. Then the
event having the space-time coordinates (x1,y1,z1,t1) in system S1 is observed in the reference system S′ which moves with velocity vy (the projection of v along the Oy axis) against the reference
system S (this relative speed being measured in system S). That corresponds to a relative
speed(7.2)vy(c)=vy1−vx2/c2between the systems S and S′ (due to the kinematics law of addition of
speeds in special relativity theory). Thus will result
in  the cuadridimensional vector r′ (having the components x′,⁢y′,⁢z′,⁢ict′), measured in system S′,
under the form(7.3)r′=L(vy(c))L(vx)r.But we can also consider that
the event having the space-time coordinates x,⁢y,⁢z,⁢t in system S is first observed in a reference system S2 which moves with velocity vy (the projection of velocity v along the Oy axis) as against system S;
a set of space-time coordinates will be established for the event. Then this
event having the space-time coordinates x2,y2,⁢z2,t2 in system S2 is observed in the reference system S′ which moves with velocity vx (the projection of velocity v along the Ox axis) against the reference
system S2,
the velocity vx being measured in the reference system S.
That corresponds to a relative speed(7.4)vx(c)=vx1−vy2/c2between the systems S′ and S2 (due to the same kinematics law of addition of
speeds in special relativity). Thus will result in
the space-time coordinates x′,⁢y′,⁢z′,⁢t′ measured in system S′ under the form(7.5)r′=L(vx(c))L(vy)r.Using the explicit form of
Lorentz transformation for the case when the relative speed has the direction
of one of the axes of coordinates, it can be easy shown that(7.6)L(vy(c))L(vx)r≠L(vx(c))L(vy)r.This shows that the coordinates
measured for the event in S′ reference system depends on the succession of
transformations. This aspect is similar to the noncommutative properties of
operators in quantum theory [16]. It implies that in the case of special relativity we
must define a vector of state (a wave-function) upon which the Lorentz
transformation acts. Thus the Lorentz transformation can be considered as a
physical transformation which modifies a certain wave function inside a
reference system. Taking into account the fact that usually we receive
information under the form of electromagnetic (or light) wave trains (the
emission of these wave trains corresponding to the event) and taking also into
account the fact that the time-dilation phenomenon (a consequence of Lorentz
transformation) was first time observed for light wave trains (the transverse
Doppler effect) it results that in the most general case this wave function
must be associated to the wave-function of the received light wave train. As a
consequence of the previous statement, it results that a Lorentz transformation L must be always put in correspondence with a
pair (S,φ), S representing a certain material reference
system which acts upon a wave train having the state-vector φ. So the Lorentz transformation must be
written under the form LS(φ);
in the most general case L is the Lorentz matrix and φ is a vector or a higher-order tensor which
describes the field. For an electromagnetic wave, the field can be described
using the cuadridimensional vector A.
The action of the matrix LS consists in a general transformation(7.7)φ(x,y,z,t)→φ′(x′,y′,z′,t′)=LSφ(x,y,z,t),where the values of φ are modified according to the transformation
rules of vectors and tensors (e.g., A′=LA for an electromagnetic wave described by the
cuadrivector A) and in the change of the space-time
coordinates (x,y,z,t) into (x′,y′,z′,t′) according to the formula(7.8)r′=LSr,r representing the cuadridimensional vector of
coordinates. We have to point the fact that in all these formulae φ(x,y,z,t) represents the value φ would have possessed in the absence of the
interaction with the observer material medium; the space-time origin must be
considered in the point of space and at the moment of time where the wave first
time interacts with the observer material medium (in a similar way with the
aspects in quantum mechanics, where all transformations are acting after the
interaction with the measuring system). This interpretation can solve the
contradictions appearing in case of movements on closed-loop trajectories (the
twins paradox) in a very simple manner. The Lorentz transformation being a
transformation which acts upon a certain wave train (a light wave train, in the
most general case), it has no consequences upon the age of two observers moving
on closed-loop trajectories. So no contradiction can appear when the two
observers are meeting again.


### 7.2. Possibilities of Using the Principle of Least Action in Connection with the Wave-Train Interpretation

We begin by writing the propagation equation for an
electromagnetic wave inside an observer material medium under the formdx2+dy2+dz2=c2dt2 (c representing the light speed). It results that c2dt2−dx2−dy2−dz2=0 for all points inside the material medium
where the wave has arrived. But(7.9)c2dt2−dx2−dy2−dz2=ds2,where ds is the cuadridimensional space-time interval.
The propagation equation of the optical wave can be written as ds=0,
and so it results that the trajectory of the wave inside the material medium
between two points a and b is determined by the equation(7.10)∫abds=Δs=0.By the other hand, for
mechanical phenomena the quantity determining the trajectory of a material body
inside a reference system is the action S.
Under a relativistic form, it can be written as S=−mc∫abds, m representing the mass of the body, and a,b the space-time coordinates for two points
situated along the “universe line” on which the body moves. The
principle of least action can be written as δS=−mcδ∫abds=0.WhileδS=∑imcuiδxi,
where ui=vi/1−v2/c2 for i=1,2,3 and u4=ic/1−v2/c2, it results finally that ∑ipi2=−m2c2, pi being the cuadrivector ∂S/∂xi (the momentum). For a free particle, pi=mui.
It can be noticed that the infinite small cuadridimensional interval ds is used both for describing the propagation of
an electromagnetic wave and the movement of a body inside a reference system.
While is it related to the action S,
this result is easy to be understood (the principle of least action being a
basic principle in nature). The next step consists in pointing the fact that
the previous integral Δs=0 (determining the trajectory of the optical
wave train inside the material medium) is based on the supposition that both
points a,⁢b belong to the material medium (otherwise, the
velocity of the wave may differ, depending on the dielectric and magnetic
constants of the material). So the equation can be directly used in measurement
procedures (for establishing trajectory or other properties of the wave only
for the time interval when the optical wave train exists in that material
medium [17]). If an
observer has to analyze a wave train emitted in another material reference
system, he must use the invariance property of the cuadriinterval: ds=ds′,
where ds represents the cuadriinterval between two
close events in a certain inertial reference system and ds′ represents the cuadriinterval between the same
two events measured in another reference system. While ds=ds(dx,dy,dz,dt) is determined inside the observer reference
system and ds′=ds′(dx′,dy′,dz′,dt′) corresponds to the reference system where the
wave has been emitted, it results that the cuadridimensional interval ds moves into the cuadridimensional interval ds′ by a function(7.11)ds(dx,dy,dz,dt)⇒L⇒ds′(dx′,dy′,dz′,dt′),where the arguments of ds are transformed by the Lorentz
relations(7.12)dx′=dx+vdt1−v2/c2,dy′=dy,dz′=dz,dt′=dt+vdx/c21−v2/c2for v parallel to Ox (all the space and time
intervals dx,⁢dy,⁢dz,⁢dt being considered inside the observer material
medium after the emitted optical wave train arrives), and ds=ds′.
The above relation can be considered as presenting a transformation of the
received wave train (with x,⁢y,⁢z,⁢t coordinates) into a “supposed” wave
train corresponding to the case when the wave train would not have entered
inside the observer material medium. For determining the real trajectory of the
wave before interaction the observer must extend the trajectory of the received
wave train (having coordinates x′,⁢y′,⁢z′,⁢t′ in the past and outside the observer material
medium, using the relation(7.13)∫abds′=Δs′=0.


### 7.3. Non-Markov Aspects of Pulse Transformation

We have also to emphasize the non-Markov aspect of
Lorentz transformation which acts upon a received wave train when this
interacts with the observer material medium. At the initial moment of time (the
zero moment) we can consider that new  values for wave quantities are generated
as a result of the Lorentz matrix action upon the received values
(cuadrivectors or cuadritensors). This represents a Markov transformation
(using some physical quantities defined at a certain moment of timet=0,
we can obtain the result of that transformation at a time moment t+dt=0+dt).Yet if we analyze the wave train transformation at a
subsequent moment of time (after the zero moment when the wave was received) we
can notice that the physical quantities corresponding to cuadrivectors and
cuadritensors are not just modified (by the action of Lorentz matrix) but are
also translated at a different time moment (according to Lorentz formulae for
transforming space-time coordinates). This implies that the physical quantities
corresponding to the transformed wave train (defined in the observer material
reference system) depend on the physical quantities corresponding to the
unchanged wave train (supposed situation) at a previous time moment. Not being
possible to use values of certain quantities at a time momentt for obtaining the values of that physical
quantities at a time moment t+dt for t>0,
it results that the Lorentz transformation of a received wave train (an
electromagnetic or optic pulse or an associated wave corresponding to a
particle) is a non-Markov transformation. In future studies, this aspects
should be studied using aspects connected to time series inside a material
medium [18].


## 7.1. The Necessity for Associating a Wave Function to the Lorentz Transformation

The Lorentz transformation is usually represented as a
matrixL which acts upon a quadridimensional column
vector r having the components r1=x,⁢r2=y,⁢r3=z,⁢r4=ict,
resulting in  another quadridimensional vector r*
having the components r1′=x′,⁢r2′=y′,⁢r3′=z′,⁢r4′=ict′,
where x,⁢y,⁢z,⁢t are the space-time coordinates corresponding
to a certain event in an inertial reference system S,
and x′,⁢y′,⁢z′,⁢t′ are the space-time coordinates corresponding
to the same event measured in an inertial reference system S′ which moves with velocity v  (a vector) against the system S. This
means(7.1)r′=L(v)r.All time moments are considered after a
synchronization moment (when the clock indications in the reference systems are
set to zero). The velocityv defines the matrix L,
and the result is considered not to depend on the measuring method used. But
let us consider that the velocity v has two components vx and vy oriented along the Ox axis (for vx) and along the Oy axis (for vy) and let us consider also that the event
taking place in the reference system S is first observed in a reference system S1 which moves with velocity vx as against the system S
:
a set of space-time coordinates (x1,y1,z1,t1) will be established for the event. Then the
event having the space-time coordinates (x1,y1,z1,t1) in system S1 is observed in the reference system S′ which moves with velocity vy (the projection of v along the Oy axis) against the reference
system S (this relative speed being measured in system S). That corresponds to a relative
speed(7.2)vy(c)=vy1−vx2/c2between the systems S and S′ (due to the kinematics law of addition of
speeds in special relativity theory). Thus will result
in  the cuadridimensional vector r′ (having the components x′,⁢y′,⁢z′,⁢ict′), measured in system S′,
under the form(7.3)r′=L(vy(c))L(vx)r.But we can also consider that
the event having the space-time coordinates x,⁢y,⁢z,⁢t in system S is first observed in a reference system S2 which moves with velocity vy (the projection of velocity v along the Oy axis) as against system S;
a set of space-time coordinates will be established for the event. Then this
event having the space-time coordinates x2,y2,⁢z2,t2 in system S2 is observed in the reference system S′ which moves with velocity vx (the projection of velocity v along the Ox axis) against the reference
system S2,
the velocity vx being measured in the reference system S.
That corresponds to a relative speed(7.4)vx(c)=vx1−vy2/c2between the systems S′ and S2 (due to the same kinematics law of addition of
speeds in special relativity). Thus will result in
the space-time coordinates x′,⁢y′,⁢z′,⁢t′ measured in system S′ under the form(7.5)r′=L(vx(c))L(vy)r.Using the explicit form of
Lorentz transformation for the case when the relative speed has the direction
of one of the axes of coordinates, it can be easy shown that(7.6)L(vy(c))L(vx)r≠L(vx(c))L(vy)r.This shows that the coordinates
measured for the event in S′ reference system depends on the succession of
transformations. This aspect is similar to the noncommutative properties of
operators in quantum theory [16]. It implies that in the case of special relativity we
must define a vector of state (a wave-function) upon which the Lorentz
transformation acts. Thus the Lorentz transformation can be considered as a
physical transformation which modifies a certain wave function inside a
reference system. Taking into account the fact that usually we receive
information under the form of electromagnetic (or light) wave trains (the
emission of these wave trains corresponding to the event) and taking also into
account the fact that the time-dilation phenomenon (a consequence of Lorentz
transformation) was first time observed for light wave trains (the transverse
Doppler effect) it results that in the most general case this wave function
must be associated to the wave-function of the received light wave train. As a
consequence of the previous statement, it results that a Lorentz transformation L must be always put in correspondence with a
pair (S,φ), S representing a certain material reference
system which acts upon a wave train having the state-vector φ. So the Lorentz transformation must be
written under the form LS(φ);
in the most general case L is the Lorentz matrix and φ is a vector or a higher-order tensor which
describes the field. For an electromagnetic wave, the field can be described
using the cuadridimensional vector A.
The action of the matrix LS consists in a general transformation(7.7)φ(x,y,z,t)→φ′(x′,y′,z′,t′)=LSφ(x,y,z,t),where the values of φ are modified according to the transformation
rules of vectors and tensors (e.g., A′=LA for an electromagnetic wave described by the
cuadrivector A) and in the change of the space-time
coordinates (x,y,z,t) into (x′,y′,z′,t′) according to the formula(7.8)r′=LSr,r representing the cuadridimensional vector of
coordinates. We have to point the fact that in all these formulae φ(x,y,z,t) represents the value φ would have possessed in the absence of the
interaction with the observer material medium; the space-time origin must be
considered in the point of space and at the moment of time where the wave first
time interacts with the observer material medium (in a similar way with the
aspects in quantum mechanics, where all transformations are acting after the
interaction with the measuring system). This interpretation can solve the
contradictions appearing in case of movements on closed-loop trajectories (the
twins paradox) in a very simple manner. The Lorentz transformation being a
transformation which acts upon a certain wave train (a light wave train, in the
most general case), it has no consequences upon the age of two observers moving
on closed-loop trajectories. So no contradiction can appear when the two
observers are meeting again.


## 7.2. Possibilities of Using the Principle of Least Action in Connection with the Wave-Train Interpretation

We begin by writing the propagation equation for an
electromagnetic wave inside an observer material medium under the formdx2+dy2+dz2=c2dt2 (c representing the light speed). It results that c2dt2−dx2−dy2−dz2=0 for all points inside the material medium
where the wave has arrived. But(7.9)c2dt2−dx2−dy2−dz2=ds2,where ds is the cuadridimensional space-time interval.
The propagation equation of the optical wave can be written as ds=0,
and so it results that the trajectory of the wave inside the material medium
between two points a and b is determined by the equation(7.10)∫abds=Δs=0.By the other hand, for
mechanical phenomena the quantity determining the trajectory of a material body
inside a reference system is the action S.
Under a relativistic form, it can be written as S=−mc∫abds, m representing the mass of the body, and a,b the space-time coordinates for two points
situated along the “universe line” on which the body moves. The
principle of least action can be written as δS=−mcδ∫abds=0.WhileδS=∑imcuiδxi,
where ui=vi/1−v2/c2 for i=1,2,3 and u4=ic/1−v2/c2, it results finally that ∑ipi2=−m2c2, pi being the cuadrivector ∂S/∂xi (the momentum). For a free particle, pi=mui.
It can be noticed that the infinite small cuadridimensional interval ds is used both for describing the propagation of
an electromagnetic wave and the movement of a body inside a reference system.
While is it related to the action S,
this result is easy to be understood (the principle of least action being a
basic principle in nature). The next step consists in pointing the fact that
the previous integral Δs=0 (determining the trajectory of the optical
wave train inside the material medium) is based on the supposition that both
points a,⁢b belong to the material medium (otherwise, the
velocity of the wave may differ, depending on the dielectric and magnetic
constants of the material). So the equation can be directly used in measurement
procedures (for establishing trajectory or other properties of the wave only
for the time interval when the optical wave train exists in that material
medium [17]). If an
observer has to analyze a wave train emitted in another material reference
system, he must use the invariance property of the cuadriinterval: ds=ds′,
where ds represents the cuadriinterval between two
close events in a certain inertial reference system and ds′ represents the cuadriinterval between the same
two events measured in another reference system. While ds=ds(dx,dy,dz,dt) is determined inside the observer reference
system and ds′=ds′(dx′,dy′,dz′,dt′) corresponds to the reference system where the
wave has been emitted, it results that the cuadridimensional interval ds moves into the cuadridimensional interval ds′ by a function(7.11)ds(dx,dy,dz,dt)⇒L⇒ds′(dx′,dy′,dz′,dt′),where the arguments of ds are transformed by the Lorentz
relations(7.12)dx′=dx+vdt1−v2/c2,dy′=dy,dz′=dz,dt′=dt+vdx/c21−v2/c2for v parallel to Ox (all the space and time
intervals dx,⁢dy,⁢dz,⁢dt being considered inside the observer material
medium after the emitted optical wave train arrives), and ds=ds′.
The above relation can be considered as presenting a transformation of the
received wave train (with x,⁢y,⁢z,⁢t coordinates) into a “supposed” wave
train corresponding to the case when the wave train would not have entered
inside the observer material medium. For determining the real trajectory of the
wave before interaction the observer must extend the trajectory of the received
wave train (having coordinates x′,⁢y′,⁢z′,⁢t′ in the past and outside the observer material
medium, using the relation(7.13)∫abds′=Δs′=0.


## 7.3. Non-Markov Aspects of Pulse Transformation

We have also to emphasize the non-Markov aspect of
Lorentz transformation which acts upon a received wave train when this
interacts with the observer material medium. At the initial moment of time (the
zero moment) we can consider that new  values for wave quantities are generated
as a result of the Lorentz matrix action upon the received values
(cuadrivectors or cuadritensors). This represents a Markov transformation
(using some physical quantities defined at a certain moment of timet=0,
we can obtain the result of that transformation at a time moment t+dt=0+dt).Yet if we analyze the wave train transformation at a
subsequent moment of time (after the zero moment when the wave was received) we
can notice that the physical quantities corresponding to cuadrivectors and
cuadritensors are not just modified (by the action of Lorentz matrix) but are
also translated at a different time moment (according to Lorentz formulae for
transforming space-time coordinates). This implies that the physical quantities
corresponding to the transformed wave train (defined in the observer material
reference system) depend on the physical quantities corresponding to the
unchanged wave train (supposed situation) at a previous time moment. Not being
possible to use values of certain quantities at a time momentt for obtaining the values of that physical
quantities at a time moment t+dt for t>0,
it results that the Lorentz transformation of a received wave train (an
electromagnetic or optic pulse or an associated wave corresponding to a
particle) is a non-Markov transformation. In future studies, this aspects
should be studied using aspects connected to time series inside a material
medium [18].


## 8. Conclusions

This study has shown that certain intuitive problems
connected with measurements of sequences of pulses on closed-loop trajectories
in special relativity and noncommutative properties of operators in quantum
physics imply a more rigorous definition of measurement method and of the
interaction phenomena (classified according to a possible memory of previous
measurements), so as to avoid logical contradictions due to a possible
resynchronization. It is also shown that the use of the least action principle
requires a specific space-time interval available for a space-time measurement
in an implicit form. Due to this, it results in a certain distinction between
the set of existing space-time intervals (which can be defined on unlimited
space-time intervals) and the set of measured space-time intervals (established
using measuring methods based on waves and always defined on limited space-time
intervals).



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*Source: 410156-2008-09-01.xml*