Research Article Wavelets-Computational Aspects of Sterian ...Approach to Uncertainty Principle in High Energy Physics: A Transient Approach CristianToma Faculty of Applied Sciences,

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2013 Article ID 735452 6 pageshttpdxdoiorg1011552013735452

Research ArticleWavelets-Computational Aspects of Sterian RealisticApproach to Uncertainty Principle in High Energy PhysicsA Transient Approach

Cristian Toma

Faculty of Applied Sciences Politehnica University 313 Splaiul Independentei 060042 Bucharest Romania

Correspondence should be addressed to Cristian Toma cgtomaphysicspubro

Received 13 August 2013 Accepted 21 August 2013

Academic Editor Carlo Cattani

Copyright copy 2013 Cristian TomaThis is an open access article distributed under theCreative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This study presents wavelets-computational aspects of Sterian-realistic approach to uncertainty principle in high energy physicsAccording to this approach one cannot make a device for the simultaneous measuring of the canonical conjugate variables inreciprocal Fourier spaces However such aspects regarding the use of conjugate Fourier spaces can be also noticed in quantumfield theory where the position representation of a quantum wave is replaced by momentum representation before computing theinteraction in a certain point of space at a certainmoment of time For this reason certain properties regarding the switch from onerepresentation to another in these conjugate Fourier spaces should be established It is shown that the best results can be obtainedusing wavelets aspects and support macroscopic functions for computing (i) wave-train nonlinear relativistic transformation (ii)reflectionrefraction with a constant shift (iii) diffraction considered as interaction with a null phase shift without annihilation ofassociated wave (iv) deflection by external electromagnetic fields without phase loss and (v) annihilation of associated wave-trainthrough fast and spatially extended phenomena according to uncertainty principle

1 Introduction

According to the Sterian realistic approach [1] one cannotmake a device for the simultaneous measuring of the canon-ical conjugate variables in the conjugate Fourier spaces

Generally Heisenbergrsquos uncertainty principle states thatincompatible dynamic variables in relation to the measuringprocess satisfy the following relations

Δ119909Δ119901119909ge

ℎ

2

Δ119910Δ119901119910ge

ℎ

2

Δ119911Δ119901119911ge

ℎ

2

Δ119905Δ119864 geℎ

2

(1)

where Δ119909 Δ119910 and Δ119911 are the uncertainties in position foreach spatial axis Δ119901

119909 Δ119901119910 and Δ119901

119911are the uncertainties

in momentum and ℎ is the reduced Planck constant thus

leading to the following statement The product of the inac-curacies arising from simultaneous determination of twocanonical conjugate variables is of the order of magnitude ofPlanckrsquos constant (see [2] for more details)

These relations can be written also in the form of com-mutation relationships between corresponding incompatibleobservables according to the theorem being given two her-mitic operators 119860 and 119861 and their commutator 119862 one candemonstrate the relationship

Δ119860Δ119861 ge

⟨119862⟩

2 (2)

where the brackets for119862 indicate an expectation value (aspectemphasized in [1]) By this moment no actual physical mea-surements can avoid the limitations of these relationships

The Sterian realistic approach is based on the analysis of awave packet (the usual mathematical model for propagatingassociated wave function) From the classical theory of wavepropagation it is known that the width of a wave packetΔ119909 involves a certain width Δ119896 in the reciprocal Fourier

2 Advances in High Energy Physics

transform space which corresponds to the wave vector 119896Standard Fourier analysis involves

Δ119909Δ119896 ge 2120587 (3)

Thus uncertainties for Δ119909 and Δ119896 are interconnected Sincemomentum 119901 for a quantum particle is proportional to thewave vector of the associated wave (119901 = ℎ119896) it results in thata lower value Δ119909 involves a greater value Δ119901 for uncertaintyupon momentum Similarly for finite duration disturbancesthe standard Fourier analysis involves

Δ119905Δ120596 ge 2120587 (4)

against reciprocal Fourier transform spaces of a pair ofsignals where Δ120596 corresponds to the spectral width

Further it is considered that in quantum physics simul-taneous measurements of the canonical conjugate variables(as space momentum time energy) cannot be performedbecause these variables correspond to the reciprocal Fourierspaces The time interval required by any device for per-forming any measurement upon a certain quantum particleor system cannot be avoided The standard experiments arenot meant for simultaneous measurements of position andmomentum of a quantum system as it is usually admit-ted being a consequence of the used measurement device(Fourier transformer) through which the signals pass havinga finite speed Any quantum system is subject to uncertaintyrelations proving its dual nature Due to the requirements ofthe principle of causality in the theory of relativity one cannotmake a device for simultaneous measuring of the canonicalconjugate variables in the conjugate Fourier spaces Due tofinite speed of propagation of interactions signal switch-ing within physical system to perform Fourier transformhas a finite duration Moreover uncertainty relations areconsidered to confirm the principle of causality due to thefinite speed of propagation of interactions required by anymeasurement process

However this approach is far from being rigorous Spe-cific aspects regarding relativistic transformation of wavesimplied by measuring procedures should be added Next arigorous analysis for coherence aspects implied by differentinteraction phenomena should be performed (the Fouriertransformation being just a mathematical tool for analyzingthis coherence for wave packets corresponding to a certainquantumparticle) Finally aspects regarding creation annihi-lation of quantum particles during the measurement processshould be analyzed by taking into consideration correlationaspects in quantum field theory and the change from coor-dinate space to momentum space (reciprocal Fourier spaces)required by the mathematical model

2 Supplementary Aspects regardingRelativistic Wavelets Transformation

As has been shown in [3] a certain wave function receivedby a reference system 119878 (represented by a material medium)is transformed according to Lorentz transformation as

1206011015840

(1199091015840

1199101015840

1199111015840

1199051015840

) = 119871120601 (119909 119910 119911 119905) (5)

where the following hold

(i) The space-time coordinates 119909 119910 119911 and 119905 correspond-ing to the received wave are transformed into the 119909

10158401199101015840 1199111015840 and 119905

1015840 coordinates of the transformed waveaccording to the action of the Lorentz transformationmatrix 119871 upon the cuadrivector [119909 119910 119911 119894119905]

119879 of thissupposed coordinates119909119910 119911 and 119905 (the coordinates thewave would have had in the absence of interaction)with the space-time origin considered in the point ofspace and at the moment of time where the receivedwave first time interacts with the observerrsquos materialmedium (in fact 119871 matrix multiplies the columncuadrivector [119909 119910 119911 119894119905]119879 so as to result in the columncuadrivector [1199091015840 1199101015840 1199111015840 1199051015840]119879)

(ii) The transformed wave function 1206011015840 is represented by

a vector or a higher-order tensor which describesthe quantum fieldparticle For an electromagneticwave 120601 corresponds to the cuadridimensional vector[119860 119894119881]

119879 In the most general case 120601 corresponds toa state vector describing the quantum particle TheLorentzmatrix119871multiplies the vector of higher-ordertensor 120601 of the received wave so as to result in thevector or higher-order tensor 120601

1015840 of the transformedwave

As a consequence each Lorentz transformation is specificto a certain wave train with the zero moment of time con-sidered when the received wave first time interacts with theobserverrsquos material medium Thus this relativistic transfor-mation is connected to transient phenomena (as the propa-gation of associated waves) and nomemory of previousmea-surements is involved regarding space-time measurementsfor events in different reference (material) systems Logicalcontradictions as clock paradox do not appear any more

This aspect implies a very important property of mea-suring device to be added to Sterian realistic approach withsignificant consequences upon computational methods theobserverrsquos material medium acts in a nonlinear manner upona superposition of received wave trains 120601

119894in a certain area

Each wave train has its own amplitudes frequency and wavevector but it has also its own zero moment of time to beconsidered within Lorentz transformation Thus at a certainmoment of time there will be different time intervals Δ119905

119894for

each wave train considered from each specific time origin 1198790119894

as

Δ119905119894= 119879 minus 119879

0119894 (6)

According to Lorentz transformation of space-time coordi-nates the time interval Δ119905

1015840

119894for the wave train transformed by

the observerrsquosmaterialmedium (considered from this specifictime origin 119879

0119894) will be

Δ1199051015840

119894=

1

radic1 minus (V119888)2Δ119905119894 (7)

Thus all parts of the wave trains 120601119894received by the observerrsquos

material medium at a certain moment of time 119879 will betranslated in time with different values depending on these

Advances in High Energy Physics 3

time differences Δ1199051015840

119894considered from different time origins

1198790119894as

120601119894(119879) 997888rarr 120601

119894(1198790119894+ Δ1199051015840

119894) (8)

(a certain delay time there is no possible anticipation of afuture event) This translation is a nonlinear transformationsince parts of the reviewed wave train are translated differ-ently After this translation is performed it can be consideredthat the Lorentz matrix [119871] acts in a linear manner upon thevector or higher-order tensor specific to the wave train 120601

119894so

as to result in the final transformed wave train 1206011015840

119894as

1206011015840

119894(1198790119894+ Δ1199051015840

119894) = [119871] 120601

119894(1198790119894+ Δ1199051015840

119894) (9)

Thus it results in that high energy (relativistic) correctionsregarding finite speed for propagation of interaction for mea-surements based on quantum aspects (presented in Sterianrealistic approach) should be completed with considerationsabout nonlinear transformations for superposition of wavetrains (as shown above) so as to result in a rigorous andcomplete transient approach With each wave train beingdescribed by specific frequency and wave vector it results inthat temporal and spatial correlations are involved in theserelativistic transformations for part of wave trains Supple-mentary aspects regarding phase will be presented in the nextparagraph

3 Aspects regarding Phase Changes forSterian Realistic Approach

Themost usual transformations performed by a certainmate-rial medium upon a wave function are reflection and refrac-tion A preliminary analysis of reflectionrefraction phe-nomena is based on classical electromagnetic field whichcorresponds in fact to the wave function associated with aphoton (the electric field E magnetic field B vector potentialA and scalar potential V are the main quantities used)This wave function can explain basic aspects in wave theoryof light as reflectionrefraction angles and the influenceof polarization upon electric and magnetic fields for bothreflected and refracted beams

At first view this classical model is just an approximationsince according to the rigorous quantum theory the energycan be transmitted just in a discontinuous manner withamounts of ℎ120596 Parts of the incident (received) wave undergoan interaction with a certain interface so quantum aspectsshould be involved However as was shown in [4] an anal-ysis based exclusively on standard quantum aspects cannotexplain important phase aspects Within such a model lightconsists of photonswhich are packets of energy that primarilyinteract with interface atoms Through this interaction theenergy of the photon is absorbed by collectivised electronsof the solid crystalline lattice and the photon ceases to existThen the electron will return to a lower energy state by emit-ting a photon Each photon behaves more like a point sourceas if the light was originating right there These emittedspherical waves generate the total wavefront as the envelopethat encloses all these point-source waves

The effect of interface nonuniformities could be consid-ered as vanishing by drawing a tangent line as a global approx-imation through the front surface for each point-sourcewaveYet there is no valid argument regarding a minimum valuefor the radii of curvature of this tangent line Theoretically itcould be very small and thus the global tangent line couldconsist of a lot of local curves with significant curvatureswhich are joined together In this way a lot of divergent lightbeams could be created along the reflectedrefracted trajec-tory and the directionality would be lost very quickly

A more rigorous standard quantummodel considers thatphotons interact with collectivised electrons of the solidcrystalline lattice before being reemitted Since the associatedwave function for the collectivised electrons is representedin position for large space intervals the influence of localnonuniformities is decreasedThus a tangent line local radiusof curvature greater than a certain value can be drawn anda better directionality for reflectedrefracted wave can beobtained

However this standard quantummodel does not take intoaccount the phase shift between the incident and the reemit-ted wave for different points of the interface A completeanalysis based on quantum theory should consider that wavesreemitted from different points of the interface are part of thewavet rain corresponding to a certain photon with the prob-ability of detecting a reflectedrefracted photon being deter-mined by the coherent plane-wave compounding method (itis well known that a particle interferes just with itself) Thereis no valid argument based exclusively on standard quantumtheory regarding the constant phase shift between the localincident wave and the corresponding local reemitted wavein each interface point The wave function for collectivisedelectrons of the crystal lattice is far from being constant inspace time along this interface

For this reason space correlations for the incident wavetrain and the reemitted wave train should be taken intoaccount This could suggest that lattice quantum functionsinteract in a global manner with parts of the incident wave ona large spatial area of the interface with a certain correlationbetween phases of reflectedrefracted wave being noticedeven for surface points separated by crystal defects Somesupport functions generated on this interface able to correlatethe phase of reemitted waves by remote interface points couldbe taken into consideration for a complete computationalmodel

In the same study [4] showed that space correlationsachieved within a very short time interval for nonadjacentspatial intervals which interact with wavefronts (part of thereflectedrefracted wave) become a key issue within an exclu-sively quantum model if we consider that reflectedrefractedwave can undergo diffraction at a later timeThe requirementof constant phase shift for parts of the associated wavegenerated by points or edges of a diffraction grating is stillvalidmdashyet the points or edges of such a diffraction grating aresituated on nonadjacent areas so they cannot be correlated byany surface quantum wave functions

These aspects imply either the use of space-propagationproperties for the support functions previously mentionedor the use of the assumption that the reemitted wavefronts


are in phase with the received wavefronts (parts of the sameincident wave) with the phase shift being null anywhere atany time

The second choice is far more attractive (being moresimple and connected to quantum field theory) However itimplies the extension of phase conservation from receptionemission of associatedwaves to receptionemission of parts ofthe same associated wave Moreover it requires a distinctionbetween these reflectionrefraction or diffraction phenomena(which does not alter coherence but modifies momentum)and the annihilationcreation phenomena involved by anyinteraction in quantum field theory otherwise any reflec-tionrefraction or diffraction of a part of the incident wave(considered as a sequence annihilation-creation) would gen-erate the annihilation of the entire associatedwave function inany other point of space (the wave function vanishes instan-taneously)

However the use of some support functions for com-putational aspects (similar to wavelets presented in [5 6])is still required in certain circumstances usually when theproperties of the material medium imply a nonzero phaseshift (the same in any point where the wave function interactswith it) An example is represented by the 120587 phase shift(corresponding to 12 of the wavelength) for reflection onmetallic surfaces when the electric field E should vanish veryquickly in any point of themetallic surface such that all partsof the wave function are to be shifted by 120587 before spatialreflection

For this reason a complete computational model cannotbe based exclusively on standard quantum aspects Certainsupport functions corresponding to spatial coherence onextended space intervals should be added for a correct com-putation ofwave trajectories in case of reflectionrefraction ordiffraction phenomena Moreover the assumption regardingnonannihilation of the entire wave function when certainparts of it undergo interaction with phase conservation (oreven constant phase shift) should be also added for a completemodelThese phase aspects should be added to Sterian realis-tic approach for correct determination of wave space proper-ties (as the wave vector or the momentum of the associatedwave) in a more general and rigorous transient approachAt this moment Sterian realistic approach uses just spatialselection for computing thewave vector (a certain direction isselected) without taking into consideration phase properties

4 Implications of Using Quantum FieldTheory upon the Realistic Approach

Any measurement for the wave vector (proportional to themomentum of the quantum particle) cannot be based exclu-sively on the spatial selection of certain directions (as isconsidered in the Sterian realistic approach in an implicitmanner due to the use of Fourier space) Any direction ofpropagation generated by a diffraction grating for exampleimplies a validation by means of a final interaction with thematerialmedium (the particle is annihilatedwhile it interactsgenerating a specific signal for the measurement device)

First wemust take into account the possibility for the par-ticle to be deflected or captured (for a very short time interval)

by the electric or magnetic field of the measurement deviceAs was shown in [7] the standard second quantificationcannot describe these interactions without the use of certainsupport functions The use of virtual photons for analyzingthe trajectory of an electron generates the phenomenon ofphase loss due to multiple interactions to be computed alongthe trajectory For example an electrostatic field should bedecomposed (using the Fourier transformation) in a set ofwaves with a certain angular frequency 120596 and a certain wavevector 119896

119860119890

120583(1199090 ) =

1

(2120587)2int119889119902119860

119890

120583(119902) 119890minus119894119902119909

(10)

where the product 119902119909 stands for

119902119909 = 11990201199090minus 119902 (11)

with1199090representing the time coordinate = (119909

1 1199092 1199093) rep-

resenting the vector of position 1199020representing the angular

frequency and 119902 = (1199021 1199022 1199023) representing the wave vector

The measurement system is chosen so as 119888 = 1 ℎ(2120587) =

1 for performing a better correspondence from the angularfrequency to the energy and from the wave vector to themomentumof the quantumparticle (see [8] formore details)

The quantities 119860119890

120583(119909) and 119860

119890

120583(119901) do not correspond to

standard photons and for this reason they cannot be sub-stituted by operators as required by second quantificationtheory However experimental facts have shown that theelectromagnetic field effect can be studied using the pertur-bations method from quantum theory

Using this method [8] the matrix element (for the firstorder of perturbation) corresponding to the electromagneticinteraction between the electron and the electromagneticfield is represented by

119878(1)

119891119894

= minus119894119890 int 119889119909V(+)1199031015840

(119891) 119890119894119901119891119909

int119889119902119860119890

120583(119902) 120574120583119890minus119894119902119909V(minus)119903

(119894) 119890minus119894119901119894119909

(12)

where V(plusmn)119903

stands for the basic vectors of an electron withpositive or negative frequencies (energies)mdashusually columnvectors V(plusmn)

119903stands for their Dirac conjugate vectorsmdashusually

line vectors index 119903 denotes the projection of the spin along acertain axis and

119891and

119894correspond to the final and initial

three-dimensional momenta for the electron respectivelyThese can be also written as

119878(1)

119891119894= minus119894119890 int 119889119902V(+)

1199031015840

(119891) 120574120583119860119890

120583(119902) V(minus)119903

(119894) 120575 (119902 minus 119901

119891+ 119901119894)

(13)

In both previous equations the expression 119860119890

120583120574120583corre-

sponds to the sum

119860119890

120583120574120583= 119860119890

01205740+

3

sum

119899=1

119860119890

119899120574119899 (14)


where 119860119890

120583represents the cuadrivector of the electromagnetic

field and 120574120583represents the Dirac matrices (its argument cor-

responds to the energy and momentum conservation laws)If the electromagnetic field does not depend on time the

cuadripotential 119860119890120583(119909) can be presented as

119860119890

120583(119909) = 119860

119890

120583() =

1

(2120587)32

int119889119902119890119894 119902 119909

119860119890

120583( 119902) (15)

which shows that the virtual photons composing the electro-magnetic field have a nonzero value just for the momentum(the energy corresponding to quantity 119902

0is equal to zero)

As a consequence the matrix element for the first order ofperturbation can be written as

119878(1)

119891119894= minus119894119890 int 119889119902V(+)

1199031015840

(119891) 120574120583119860119890

120583( 119902) V(minus)119903

(119894)

times 120575 ( 119902 minus 119891+ 119894) 120575 (119901

1198910minus 1199011198940)

(16)

By performing the integration on 119889119902 it results in

119878(1)

119891119894= minus119894119890V(+)

1199031015840

(119891) 120574120583119860119890

120583( 119902) V(minus)119903

(119894) 120575 (119901

1198910minus 1199011198940) (17)

According to standard interpretation of quantum field the-ory this first order element from perturbation method isconnected to the probability of an interaction between anelectron with initial momentum

119894and energy 119901

1198940and a

virtual photonwithmomentum 119902 so as to result in an electronwith momentum

119891and energy 119901

1198910

However for computing the whole interaction for anelectron in an electrostatic field (so as to determine thetrajectory of the associated wave train) we have to considerthat the electron has to undergo multiple interactions withsuch virtual photons until the action of the exterior fieldvanishes (such multiple interactions are allowed by secondquantification theory) According to the quantum laws eachinteraction transforms the initial electron (its wave trainused in computation) into a new wave train (the final elec-tron) with different characteristics (another momentum andenergy) and so on

Considering that the wave train corresponding to theelectron undergoes a set of interactions with virtual photonsit results in that a certain transient time is required byeach interaction so as physical quantities as wavelength andangular frequency to be transferred in this local phenomenon(these quantities are used for further computing the matrixelements for creationannihilation phenomena) This tran-sient time causes a phase loss for the initial wave train soit results in that the great number of interactions of theinitial electron in an electrostatic (Coulombian) field wouldfinally cause significant phase loss (the time length of thewave train associated with the final electron tends to zero)This corresponds to a vanishing phenomenon for the initialelectron in contradiction with experimental facts

This phenomenon can be avoided into a complete and rig-orous transient approach if the whole trajectory is computedusing just one Lagrangian function for the whole interaction

of the electron with the electromagnetic field according tothe dynamical equations

(minus1205972

1205971199052+ nabla2

)119860120583(119909) = minus119890120595120574

120583120595

(119894120574120583

120597

120597119909120583

minus 119890120574120583119860120583)120595 (119909) minus 119898120595 (119909) = 0

120595 (119909) (119894120574120583

120597

120597119909120583

+ 119890120574120583119860120583) + 119898120595 (119909) = 0

(18)

(see [8]) Thus the wave function corresponding to the elec-tron is modified in a continuous manner along its trajectory(according to the first quantification theory) The exteriorfield acts as a support function (as mentioned in previousparagraph) determining the evolution of the electron wavetrain Moreover it shows the need for a certain reference sys-tem which acts upon the wave trains corresponding toquantum particles and their associated waves in any rigorouscomputational model for transient phenomena (the exteriorelectromagnetic field is part of it) This aspect is supportedby the higher magnitude and a slow time variation of theseexterior electromagnetic fields and by the lack of reversibilityfor diffraction phenomena in quantumphysics (we can noticean electron diffraction phenomenon when an electron beaminteracts with a motionless crystal lattice but we cannotimagine a diffraction of an entire crystal lattice when it inter-acts with a motionless spatial distribution of electrons)

Similar to reflectionrefraction and diffraction phenom-ena previously presented these deflections do not alter coher-enceThey just alter the spatial directions (the wave vector) ofcertain wave trains previously selected (eg by a diffractiongrating) After corrections due to possible deflections arecomputed we should analyze the annihilation phenomenonwhich validates the detection of a particle (wave train) withcertain characteristics The rigorous model is based on quan-tum field theory (no virtual particles are taken into consid-eration) According to this model the wave train is analyzedusing the momentum space The switch from position spaceto momentum space is performed using the Fourier transfor-mation The space and time origins are selected in this pointof interaction when the associated wave train is receivedby the material medium The annihilation of the particleimplies the instant annihilation of all parts of this wave trainirrespective of the distance to this interaction point It is truethat this seems to contradict the relativity postulated (nospeed can surpass the light speed) but in fact any speed can benoticed just by an emission and a reception of a certain signalor particle The annihilation of a certain quantum particlecannot be noticed in two different space areas Thus thereis no contradiction Moreover the existence of some highspeed support functions which cancel the wave in an instantmanner can be supported by uncertainty principle also Sincethe particle ceases to exist its final momentum is zero Beingno uncertainty it results in that

Δ119901119909= Δ119901119910= Δ119901119911= 0 (19)


and according to uncertainty principle

Δ119909 geℎ

Δ119901119909

997888rarr infin

Δ119910 geℎ

Δ119901119910

997888rarr infin

Δ119911 geℎ

Δ119901119911

997888rarr infin

(20)

Thus this annihilation phenomenon can be considered asbeing infinitely extended Unlike standard phenomena con-sidered by Sterian realistic approach the uncertainty formomentum determines the uncertainty for space intervalsHowever for computational aspects it can be simply noticedthat discontinuities cannot be analyzed using the differentialform of wave equations A discontinuity looks like a limita-tion but physically they are represented by fracture phenom-ena for certain quantities An avalanche process can be easilyimagined and computed (similar tomechanical engineering)

5 Conclusions

This study has presented supplementary aspects for the Ste-rian realistic approach to uncertainty principle It was shownthat high energy (relativistic) corrections regarding finitespeed for propagation of interaction for measurements basedon quantumaspects should be completedwith considerationsabout nonlinear transformations for superposition of wavetrains so as to result in a rigorous and complete transientapproach With each wave train being described by a specificfrequency wave vector and its own Lorentz transformation(with specific time origin) it results in that temporal andspatial correlations are involved in these relativistic transfor-mations for part of wave trains

It was also shown that a complete computational modelfor a transient approach cannot be based exclusively onstandard quantum aspects Certain support functions corre-sponding to spatial coherence on extended space intervals(similar to wavelets propagation in compositematerials [9] orto multiscale phenomena [10]) should be added for a correctcomputation of wave trajectories in case of reflectionrefrac-tion or diffraction phenomena Moreover an assumptionregarding nonannihilation of the entire wave function whencertain parts of it undergo interaction with phase conserva-tion (or even constant phase shift) should be also added fora complete model The need for a certain exterior referencesystem acting upon the quantum wave train is also presented(based on some diffraction aspects)

Computational aspects for deflection of associated wavesin electromagnetic fields were also studied It emphasized theneed for a certain reference system which acts upon the wavetrains corresponding to quantum particles and their associ-ated waves in any rigorous computationalmodel for transientphenomena (the exterior electromagnetic field is part ofit as support functions) so that phase-loss phenomenongenerated by the use of second quantification is to be avoided

Finally it was shown that both the switch from positionspace to momentum space and the annihilation of a certain

quantum particle are represented by instant phenomenasupported by uncertainty principle (unlike considerationsin Sterian realistic approach) since there is no uncertaintyfor final momentum when the particle is annihilated aninfinitely extended phenomenon is generated (an instantpropagating spatial noise is generated similar to temporalGaussian noisemdashsee [11] and frequency dependent noisemdashsee [12])

References

[1] P E Sterian ldquoRealistic approach of the relations of uncertaintyof Heisenbergrdquo Advances in High Energy Physics vol 2013Article ID 872507 7 pages 2013

[2] LD Landau andEM LifshitzQuantumMechanics Non-Rela-tivistic Theory vol 3 Pergamon Press Oxford UK 3rd edition1974

[3] E G Bakhoum and C Toma ldquoRelativistic short range phenom-ena and space-time aspects of pulse measurementsrdquoMathemat-ical Problems in Engineering vol 2008 Article ID 410156 20pages 2008

[4] E Bakhoum and C Toma ldquoTransient aspects of wave propaga-tion connected with spatial coherencerdquoMathematical Problemsin Engineering vol 2013 Article ID 691257 5 pages 2013

[5] C Cattani ldquoFractional calculus and Shannon waveletrdquo Mathe-matical Problems in Engineering vol 2012 Article ID 502812 26pages 2012

[6] J Leng T Huang and C Cattani ldquoConstruction of bivariatenonseparable compactly supported orthogonalwaveletsrdquoMath-ematical Problems in Engineering vol 2013 Article ID 62495711 pages 2013

[7] E G Bakhoum and C Toma ldquoMathematical transform of trav-eling-wave equations and phase aspects of quantum interac-tionrdquo Mathematical Problems in Engineering vol 2010 ArticleID 695208 15 pages 2010

[8] N Nelipa Physique des Particules Elementaires Editions MIRMoscou Russia 1981

[9] J J Rushchitsky C Cattani andEV Terletskaya ldquoWavelet anal-ysis of the evolution of a solitary wave in a composite materialrdquoInternational AppliedMechanics vol 40 no 3 pp 311ndash318 2004

[10] C Cattani ldquoMultiscale analysis of wave propagation in compos-ite materialsrdquo Mathematical Modelling and Analysis vol 8 no4 pp 267ndash282 2003

[11] M Li andW Zhao ldquoOn bandlimitedness and lag-limitedness offractional Gaussian noiserdquo Physica A vol 392 no 9 pp 1955ndash1961 2013

[12] M Li and W Zhao ldquoOn 1119891 noiserdquo Mathematical Problems inEngineering vol 2012 Article ID 673648 23 pages 2012

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transform space which corresponds to the wave vector 119896Standard Fourier analysis involves

Δ119909Δ119896 ge 2120587 (3)

Thus uncertainties for Δ119909 and Δ119896 are interconnected Sincemomentum 119901 for a quantum particle is proportional to thewave vector of the associated wave (119901 = ℎ119896) it results in thata lower value Δ119909 involves a greater value Δ119901 for uncertaintyupon momentum Similarly for finite duration disturbancesthe standard Fourier analysis involves

Δ119905Δ120596 ge 2120587 (4)

against reciprocal Fourier transform spaces of a pair ofsignals where Δ120596 corresponds to the spectral width

Further it is considered that in quantum physics simul-taneous measurements of the canonical conjugate variables(as space momentum time energy) cannot be performedbecause these variables correspond to the reciprocal Fourierspaces The time interval required by any device for per-forming any measurement upon a certain quantum particleor system cannot be avoided The standard experiments arenot meant for simultaneous measurements of position andmomentum of a quantum system as it is usually admit-ted being a consequence of the used measurement device(Fourier transformer) through which the signals pass havinga finite speed Any quantum system is subject to uncertaintyrelations proving its dual nature Due to the requirements ofthe principle of causality in the theory of relativity one cannotmake a device for simultaneous measuring of the canonicalconjugate variables in the conjugate Fourier spaces Due tofinite speed of propagation of interactions signal switch-ing within physical system to perform Fourier transformhas a finite duration Moreover uncertainty relations areconsidered to confirm the principle of causality due to thefinite speed of propagation of interactions required by anymeasurement process

However this approach is far from being rigorous Spe-cific aspects regarding relativistic transformation of wavesimplied by measuring procedures should be added Next arigorous analysis for coherence aspects implied by differentinteraction phenomena should be performed (the Fouriertransformation being just a mathematical tool for analyzingthis coherence for wave packets corresponding to a certainquantumparticle) Finally aspects regarding creation annihi-lation of quantum particles during the measurement processshould be analyzed by taking into consideration correlationaspects in quantum field theory and the change from coor-dinate space to momentum space (reciprocal Fourier spaces)required by the mathematical model

2 Supplementary Aspects regardingRelativistic Wavelets Transformation

As has been shown in [3] a certain wave function receivedby a reference system 119878 (represented by a material medium)is transformed according to Lorentz transformation as

1206011015840

(1199091015840

1199101015840

1199111015840

1199051015840

) = 119871120601 (119909 119910 119911 119905) (5)

where the following hold

(i) The space-time coordinates 119909 119910 119911 and 119905 correspond-ing to the received wave are transformed into the 119909

10158401199101015840 1199111015840 and 119905

1015840 coordinates of the transformed waveaccording to the action of the Lorentz transformationmatrix 119871 upon the cuadrivector [119909 119910 119911 119894119905]

119879 of thissupposed coordinates119909119910 119911 and 119905 (the coordinates thewave would have had in the absence of interaction)with the space-time origin considered in the point ofspace and at the moment of time where the receivedwave first time interacts with the observerrsquos materialmedium (in fact 119871 matrix multiplies the columncuadrivector [119909 119910 119911 119894119905]119879 so as to result in the columncuadrivector [1199091015840 1199101015840 1199111015840 1199051015840]119879)

(ii) The transformed wave function 1206011015840 is represented by

a vector or a higher-order tensor which describesthe quantum fieldparticle For an electromagneticwave 120601 corresponds to the cuadridimensional vector[119860 119894119881]

119879 In the most general case 120601 corresponds toa state vector describing the quantum particle TheLorentzmatrix119871multiplies the vector of higher-ordertensor 120601 of the received wave so as to result in thevector or higher-order tensor 120601

1015840 of the transformedwave

As a consequence each Lorentz transformation is specificto a certain wave train with the zero moment of time con-sidered when the received wave first time interacts with theobserverrsquos material medium Thus this relativistic transfor-mation is connected to transient phenomena (as the propa-gation of associated waves) and nomemory of previousmea-surements is involved regarding space-time measurementsfor events in different reference (material) systems Logicalcontradictions as clock paradox do not appear any more

This aspect implies a very important property of mea-suring device to be added to Sterian realistic approach withsignificant consequences upon computational methods theobserverrsquos material medium acts in a nonlinear manner upona superposition of received wave trains 120601

119894in a certain area

Each wave train has its own amplitudes frequency and wavevector but it has also its own zero moment of time to beconsidered within Lorentz transformation Thus at a certainmoment of time there will be different time intervals Δ119905

119894for

each wave train considered from each specific time origin 1198790119894

as

Δ119905119894= 119879 minus 119879

0119894 (6)

According to Lorentz transformation of space-time coordi-nates the time interval Δ119905

1015840

119894for the wave train transformed by

the observerrsquosmaterialmedium (considered from this specifictime origin 119879

0119894) will be

Δ1199051015840

119894=

1

radic1 minus (V119888)2Δ119905119894 (7)

Thus all parts of the wave trains 120601119894received by the observerrsquos

material medium at a certain moment of time 119879 will betranslated in time with different values depending on these




1198790119894as

120601119894(119879) 997888rarr 120601

119894(1198790119894+ Δ1199051015840

119894) (8)


119894so


119894as

1206011015840

119894(1198790119894+ Δ1199051015840

119894) = [119871] 120601

119894(1198790119894+ Δ1199051015840

119894) (9)




















119860119890

120583(1199090 ) =

1

(2120587)2int119889119902119860

119890

120583(119902) 119890minus119894119902119909

(10)


119902119909 = 11990201199090minus 119902 (11)


1 1199092 1199093) rep-






120583(119909) and 119860

119890




119878(1)

119891119894

= minus119894119890 int 119889119909V(+)1199031015840

(119891) 119890119894119901119891119909

int119889119902119860119890

120583(119902) 120574120583119890minus119894119902119909V(minus)119903

(119894) 119890minus119894119901119894119909

(12)





119891and



119878(1)

119891119894= minus119894119890 int 119889119902V(+)

1199031015840

(119891) 120574120583119860119890

120583(119902) V(minus)119903

(119894) 120575 (119902 minus 119901

119891+ 119901119894)

(13)


120583120574120583corre-

sponds to the sum

119860119890

120583120574120583= 119860119890

01205740+

3

sum

119899=1

119860119890

119899120574119899 (14)


where 119860119890





119860119890

120583(119909) = 119860

119890

120583() =

1

(2120587)32

int119889119902119890119894 119902 119909

119860119890

120583( 119902) (15)


0is equal to zero)


119878(1)

119891119894= minus119894119890 int 119889119902V(+)

1199031015840

(119891) 120574120583119860119890

120583( 119902) V(minus)119903

(119894)

times 120575 ( 119902 minus 119891+ 119894) 120575 (119901

1198910minus 1199011198940)

(16)


119878(1)

119891119894= minus119894119890V(+)

1199031015840

(119891) 120574120583119860119890

120583( 119902) V(minus)119903

(119894) 120575 (119901

1198910minus 1199011198940) (17)


119894and energy 119901

1198940and a


119891and energy 119901

1198910





(minus1205972

1205971199052+ nabla2

)119860120583(119909) = minus119890120595120574

120583120595

(119894120574120583

120597

120597119909120583

minus 119890120574120583119860120583)120595 (119909) minus 119898120595 (119909) = 0

120595 (119909) (119894120574120583

120597

120597119909120583

+ 119890120574120583119860120583) + 119898120595 (119909) = 0

(18)



Δ119901119909= Δ119901119910= Δ119901119911= 0 (19)



Δ119909 geℎ

Δ119901119909

997888rarr infin

Δ119910 geℎ

Δ119901119910

997888rarr infin

Δ119911 geℎ

Δ119901119911

997888rarr infin

(20)


5 Conclusions






References


















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






1198790119894as

120601119894(119879) 997888rarr 120601

119894(1198790119894+ Δ1199051015840

119894) (8)


119894so


119894as

1206011015840

119894(1198790119894+ Δ1199051015840

119894) = [119871] 120601

119894(1198790119894+ Δ1199051015840

119894) (9)




















119860119890

120583(1199090 ) =

1

(2120587)2int119889119902119860

119890

120583(119902) 119890minus119894119902119909

(10)


119902119909 = 11990201199090minus 119902 (11)


1 1199092 1199093) rep-






120583(119909) and 119860

119890




119878(1)

119891119894

= minus119894119890 int 119889119909V(+)1199031015840

(119891) 119890119894119901119891119909

int119889119902119860119890

120583(119902) 120574120583119890minus119894119902119909V(minus)119903

(119894) 119890minus119894119901119894119909

(12)





119891and



119878(1)

119891119894= minus119894119890 int 119889119902V(+)

1199031015840

(119891) 120574120583119860119890

120583(119902) V(minus)119903

(119894) 120575 (119902 minus 119901

119891+ 119901119894)

(13)


120583120574120583corre-

sponds to the sum

119860119890

120583120574120583= 119860119890

01205740+

3

sum

119899=1

119860119890

119899120574119899 (14)


where 119860119890





119860119890

120583(119909) = 119860

119890

120583() =

1

(2120587)32

int119889119902119890119894 119902 119909

119860119890

120583( 119902) (15)


0is equal to zero)


119878(1)

119891119894= minus119894119890 int 119889119902V(+)

1199031015840

(119891) 120574120583119860119890

120583( 119902) V(minus)119903

(119894)

times 120575 ( 119902 minus 119891+ 119894) 120575 (119901

1198910minus 1199011198940)

(16)


119878(1)

119891119894= minus119894119890V(+)

1199031015840

(119891) 120574120583119860119890

120583( 119902) V(minus)119903

(119894) 120575 (119901

1198910minus 1199011198940) (17)


119894and energy 119901

1198940and a


119891and energy 119901

1198910





(minus1205972

1205971199052+ nabla2

)119860120583(119909) = minus119890120595120574

120583120595

(119894120574120583

120597

120597119909120583

minus 119890120574120583119860120583)120595 (119909) minus 119898120595 (119909) = 0

120595 (119909) (119894120574120583

120597

120597119909120583

+ 119890120574120583119860120583) + 119898120595 (119909) = 0

(18)



Δ119901119909= Δ119901119910= Δ119901119911= 0 (19)



Δ119909 geℎ

Δ119901119909

997888rarr infin

Δ119910 geℎ

Δ119901119910

997888rarr infin

Δ119911 geℎ

Δ119901119911

997888rarr infin

(20)


5 Conclusions






References


















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics












119860119890

120583(1199090 ) =

1

(2120587)2int119889119902119860

119890

120583(119902) 119890minus119894119902119909

(10)


119902119909 = 11990201199090minus 119902 (11)


1 1199092 1199093) rep-






120583(119909) and 119860

119890




119878(1)

119891119894

= minus119894119890 int 119889119909V(+)1199031015840

(119891) 119890119894119901119891119909

int119889119902119860119890

120583(119902) 120574120583119890minus119894119902119909V(minus)119903

(119894) 119890minus119894119901119894119909

(12)





119891and



119878(1)

119891119894= minus119894119890 int 119889119902V(+)

1199031015840

(119891) 120574120583119860119890

120583(119902) V(minus)119903

(119894) 120575 (119902 minus 119901

119891+ 119901119894)

(13)


120583120574120583corre-

sponds to the sum

119860119890

120583120574120583= 119860119890

01205740+

3

sum

119899=1

119860119890

119899120574119899 (14)


where 119860119890





119860119890

120583(119909) = 119860

119890

120583() =

1

(2120587)32

int119889119902119890119894 119902 119909

119860119890

120583( 119902) (15)


0is equal to zero)


119878(1)

119891119894= minus119894119890 int 119889119902V(+)

1199031015840

(119891) 120574120583119860119890

120583( 119902) V(minus)119903

(119894)

times 120575 ( 119902 minus 119891+ 119894) 120575 (119901

1198910minus 1199011198940)

(16)


119878(1)

119891119894= minus119894119890V(+)

1199031015840

(119891) 120574120583119860119890

120583( 119902) V(minus)119903

(119894) 120575 (119901

1198910minus 1199011198940) (17)


119894and energy 119901

1198940and a


119891and energy 119901

1198910





(minus1205972

1205971199052+ nabla2

)119860120583(119909) = minus119890120595120574

120583120595

(119894120574120583

120597

120597119909120583

minus 119890120574120583119860120583)120595 (119909) minus 119898120595 (119909) = 0

120595 (119909) (119894120574120583

120597

120597119909120583

+ 119890120574120583119860120583) + 119898120595 (119909) = 0

(18)



Δ119901119909= Δ119901119910= Δ119901119911= 0 (19)



Δ119909 geℎ

Δ119901119909

997888rarr infin

Δ119910 geℎ

Δ119901119910

997888rarr infin

Δ119911 geℎ

Δ119901119911

997888rarr infin

(20)


5 Conclusions






References


















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




where 119860119890





119860119890

120583(119909) = 119860

119890

120583() =

1

(2120587)32

int119889119902119890119894 119902 119909

119860119890

120583( 119902) (15)


0is equal to zero)


119878(1)

119891119894= minus119894119890 int 119889119902V(+)

1199031015840

(119891) 120574120583119860119890

120583( 119902) V(minus)119903

(119894)

times 120575 ( 119902 minus 119891+ 119894) 120575 (119901

1198910minus 1199011198940)

(16)


119878(1)

119891119894= minus119894119890V(+)

1199031015840

(119891) 120574120583119860119890

120583( 119902) V(minus)119903

(119894) 120575 (119901

1198910minus 1199011198940) (17)


119894and energy 119901

1198940and a


119891and energy 119901

1198910





(minus1205972

1205971199052+ nabla2

)119860120583(119909) = minus119890120595120574

120583120595

(119894120574120583

120597

120597119909120583

minus 119890120574120583119860120583)120595 (119909) minus 119898120595 (119909) = 0

120595 (119909) (119894120574120583

120597

120597119909120583

+ 119890120574120583119860120583) + 119898120595 (119909) = 0

(18)



Δ119901119909= Δ119901119910= Δ119901119911= 0 (19)



Δ119909 geℎ

Δ119901119909

997888rarr infin

Δ119910 geℎ

Δ119901119910

997888rarr infin

Δ119911 geℎ

Δ119901119911

997888rarr infin

(20)


5 Conclusions






References


















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





Δ119909 geℎ

Δ119901119909

997888rarr infin

Δ119910 geℎ

Δ119901119910

997888rarr infin

Δ119911 geℎ

Δ119901119911

997888rarr infin

(20)


5 Conclusions






References


















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



Documents

Research Article Wavelets-Computational Aspects of Sterian ...Approach to Uncertainty Principle in High Energy Physics: A Transient Approach CristianToma Faculty of Applied Sciences,