Mathematics of Particle-Wave Mechanical Systems

388 Pages·2022·3.865 MB·English

by  James M. Hill

James M. Hill
Mathematics
of Particle-Wave
Mechanical
Systems
Mathematics of Particle-Wave Mechanical Systems
James M. Hill
Mathematics of
Particle-Wave Mechanical
Systems
JamesM.Hill
UniversityofSouthAustralia
Adelaide,SA,Australia
ISBN978-3-031-19792-5 ISBN978-3-031-19793-2 (eBook)
https://doi.org/10.1007/978-3-031-19793-2
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Author’s Foreword
Thepresenttextattemptstodevelopacombinedtheoryforthetwodistinctandwell-
defined primitive notions of particles and waves. The intended contents are most
aptly characterised in the words of the eminent Irish mathematical physicist John
LightonSynge,whoindescribingthecontentsofhisbook,GeometricalMechanics
and de Broglie Waves, writes in the introduction, ‘This book is intended as a
contributionto“mathematical”physicsratherthanto“naturalistic”physics’[100].
Bythishemeans,‘thecreationofacoherentmathematicaltheory,withclearlyset
forth assumptions being the primary object, with physical interpretation in terms
of experiment being relegated to a secondary position’ [99]. In the engineering
communitywhentheoryandobservationnolongercoincide,thenotionofa‘fudge
factor’ becomes paramount to reconcile the two positions. In the physics commu-
nity, however, the same predicament forces the introduction of new factors which
arecoinedmysteriousbecausetheycannolongerbeexplainedthroughconventional
thinking, and the word ‘dark’ becomes attached to these mysterious phenomena.
It is clear to the writer that the dark issues of astrophysics are in consequence of
incorrect mechanical accounting, namely omitting insignificant issues at a local
levelwhichbecomeincreasinglysignificantatlargerscales.Accordingly,thereisa
needtore-assessfundamentalmechanicalaccountingpracticewhileincorporating
theessentialaspectsofthesuccessfulwell-establishedareasofphysics.Thisbook
describesonesuchformulationandattemptstodevelopthekeyimplicationsofthe
proposal.
As the theory of electro-magnetism developed in the late nineteenth century,
in attempting to formulate the critical electric and magnetic concepts, pioneering
scientists such as Faraday and Maxwell sought inspiration from well-established
mechanicalanaloguesfrombothsolidandfluidmechanics.Subsequentlytherehas
always been a strong interplay between ideas and concepts in electro-magnetism
and mechanics, to the extent that the entire mechanical theory of special relativity
springsdirectlyfromLorentzinvarianceoftheequationsofelectro-magnetism.One
ofthekeylessonsinelectro-magnetismisthemorefundamentalimportanceofthe
vector and scalar potentials (A,V) as compared to the fields (E,B). The vector
potentialAcorrespondstotheelectromagneticmomentum,andistheanalogueof
vii
viii Author’sForeword
mechanicalmomentum,andreferredtobybothFaradayandMaxwellastheelectro-
tonic intensity, while the potential V is the analogue of the velocity potential in
fluid mechanics. This book adopts the perspective that a similar situation might
apply in special relativistic mechanics, and it is not just the momentum vector p
thatcontributestotheworkdoneW butrathertheparticleenergyeitselfalsoplays
an important role. We develop here a Lorentz invariant modification of Newton’s
secondlawapplyingwhenparticleenergyitselfisofcomparablemagnitudetothe
potentialenergyoftheappliedexternalfield.
While the original idea came from electro-magnetism, much later I was able
to reflect on an often-repeated remark of a colleague, who would say that general
relativity specialists do not care which one in particular of their four variables
corresponds to time in their four-dimensional description of space-time. Coming
fromcontinuummechanics,thisstruckmeasaninterestingcomment,butifitwere
true,thenwecertainlymightbewillingtoputspaceandtimeonequalfootings,and
thenotionofaforceg inthedirectionoftimemightnotbesucharadicalthought.
Indeed,attheveryheartofthetheoryofgeneralrelativityliesthenotionofanideal
test particle, the presence of which is assumed not to affect the gravitational field.
Thetheoryexpoundedhereattemptstomakesomeaccommodationforthisenergy
omissionwithinthecontextofthetheoryofspecialrelativity.
Aftertoyingwiththereplacementformulation(3.4)forNewton’ssecondlaw,the
ideaseemedtohavesomemerit.Iwasespeciallyencouragedthattheforcerelations
Eqs.(3.4)areLorentzinvariant,andthatEq.(5.8)admitsthetwosuggestivelimits
dE/dp = u and dE/dp = c2/u, where E is wave energy and p is particle
momentum.Thesetwolimitsaretheparticleandwavelimits,whichalsoarisefrom
Eq.(3.10) as purely spatial (dt = 0) or purely temporal (dx = 0) extreme limits
oftheproposedtheory.AlsoimmediatefromEq.(5.8)aretherelationsE = ±cp,
termed here the de Broglie relations. The positive case is equivalent to the known
relations for light, namely p = h/μ and E = hν, which together imply E = cp
where h is the Planck constant and μ is the wave length (c = μν). These initial
resultsprovidedsufficientincentivetocontemplateotherallowableoutcomes.
Alsoimportantisthatthestructureofthestandardoperatorrelationsofquantum
mechanics p −→ −ih¯∂/∂x and e −→ ih¯∂/∂t is immediately apparent from the
proposedtheory,assumingonlythattheexternalforcesf andgaregeneratedfrom
a potential V(x,t) such that f = −∂V/∂x and gc2 = −∂V/∂t. Specifically,
from (8.1), we have p = ∂ψ/∂x and E = ∂ψ/∂t, and conservation of energy
e+E +V =constant immediatelygivese =−(∂ψ/∂t +V),whichisprecisely
thestructureofthestandardoperatorrelationsofquantummechanics.Thatis,the
establishedstructureofconventionalquantummechanicsisimmediatelyinherentin
thepresentapproachandarisesfromconservationofenergy.
This highlights the fact that particle energy e and wave energy E must be
accommodated separately. Conventionally, energy is just energy, and in quantum
mechanics, particle-like and wave-like states are compromised by restricting the
particle-like properties so that they do not permit the description of an exact
space-time motion (Heisenberg uncertainty) achieved through the introduction of
probability waves. Here, we deal with particle energy e and wave energy E as
Author’sForeword ix
strictlydifferententities,andfortheabove-mentioned exampleoflight,wewould
say e = 0 and E = cp. The notion of energy commonly referred to as particle
energyisanacquiredcharacteristicarisingfromthepropertiesofspace,whilewave
energyisanaccumulationofenergyarisingfromtime.
Also noteworthy is the fact that a simple and logical analysis of the Einstein
energy statement e2 = e 2 +(pc)2 immediately gives rise to the prospect of four
0
distincttypesofmatter,sinceeithertherestenergye iszerooritisnon-zero,and
0
foreachofthesetwoalternatives,wemayadoptaplusorminussign.Thissimple
and logical explanation of at least four distinct types of matter is only meaningful
becauseitisinterpretedwithinthepresentproposedextensionofspecialrelativity,
and such an interpretation is simply not available within the narrower confines of
traditionalspecialrelativitybecauseitlacksthenotionofaforceinthedirectionof
time.
Here,wespeculatethatdarkmatteranddarkenergyariseasspecialorprivileged
states occurring for particular alignments of the magnitude of the spatial physical
forcef andtheforceginthedirectionoftime,whichinasinglespatialdimension
become simply f = ±gc. The particle and wave energies for these alignments
are such that e = ±E, with dark matter an essentially backward wave and dark
energyanessentiallyforwardwave,bothpropagatingatthespeedoflight.Inareal
circumstance,wemightexpectasituationcomparabletoa‘fuzzyregion’wherethe
key equalities are constantly switching on and off dependent upon a varying local
environment.Thecasee = −E isespeciallyinterestingsinceitmayoperateunder
zero potential V = 0, which might account for the abundance of dark energy and
darkmatterintheuniverse.
This text presents an elementary and much expanded introduction to the basic
mathematicsandmechanicsoftheideasformulatedin[47–52]involvingaLorentz
invariantalternativetoNewton’ssecondlaw.TheLorentzinvariantmodificationof
Newton’s second law extends conventional special relativity theory by developing
a dual particle-wave formulation accommodating both particle and wave energies,
and allowing exceptions to the law that matter cannot be created or destroyed. de
Brogliewasfirsttoproposeaconcretephysicalpictureoftheco-existenceofboth
particle and its associated wave, and [47–52] make a distinction between particle
andwaveenergieseandE,respectively,suchthatthetotalworkdonebytheparticle
W = e+E accumulates from both a spatial physical force f and a force g in the
directionoftime.Generally,experimentsareundertakenwiththeclassicalnotions
of either particles or waves in mind, so that in an experiment either particles or
wavesarereported,andonlyoneofeorE ismeasured.Sincenaturetendstoadopt
the least energy structure, for the present theory, we propose that particles appear
for e < E and waves for E < e, but in either event, both a measurable and an
unmeasurableenergyexists.
Ourspecificpurposehereistoprovideanintroductiontothemathematicaland
mechanicalframeworkunderpinningtheseideaswithalevelofmathematicaldetail
notincludedintheoriginalpapers,asmightbecomprehensibletoanundergraduate
student in either physics or applied mathematics. However, it is not a definitive
account of a successful completed story, but rather a single proposal and a log for
x Author’sForeword
some of the mathematical results pertaining to the proposed model. The applied
mathematical and mathematical modelling communities are used to the idea that
thereoftenexistarangeofperfectlysensibleandmeaningfulmathematicalmodels
predictingcomparablenumericaloutcomes.Whilethewriterisopentotheideathat
the particular model structure advanced here may not be immaculately correct, he
is nevertheless firmly of the view that the ultimate resolution of the dark issues of
mechanicswillrequireacomparablere-thinkingofNewton’ssecondlaw,andthat
the dark issues will emerge as essentially artefacts of the mechanical accounting.
The author is grateful to Professor David Steigmann for his constant support
and encouragement of this work, and to Professor Phil Broadbridge, Dr Barry
Cox, Professor Sam Drake, Dr Joe O’Leary and Mr Arthur Rorris, for numerous
enjoyablediscussionsovercoffee.
Contents
1 Introduction................................................................. 1
1.1 Introduction........................................................... 1
1.2 GeneralIntroduction ................................................. 1
1.3 SpecialRelativity..................................................... 4
1.4 QuantumMechanics ................................................. 5
1.5 deBroglieParticle-WaveMechanics................................ 7
1.6 PlanofText........................................................... 9
1.7 TablesofMajorSymbolsandBasicEquations..................... 12
2 SpecialRelativity ........................................................... 15
2.1 Introduction........................................................... 15
2.2 LorentzTransformations............................................. 16
2.3 EinsteinAdditionofVelocitiesLaw................................. 18
2.4 LorentzInvariances .................................................. 20
2.5 LorentzInvariantVelocityFieldsu(x,t)............................ 24
2.6 GeneralFrameworkforLorentzInvariances........................ 25
2.7 IntegralInvariantsoftheLorentzGroup............................ 28
2.8 AlternativeValidationofLorentzInvariants........................ 31
2.9 JacobiansoftheLorentzTransformations .......................... 32
2.10 Space-TimeTransformationx(cid:3) =ct andt(cid:3) =x/c ................. 34
2.11 ThedeBroglieWaveVelocityu(cid:3) =c2/u........................... 35
2.12 ForceandPhysicalEnergyArisingfromWorkDone.............. 37
2.13 LorentzInvariantEnergy-MomentumRelations.................... 40
2.14 ForceInvarianceforConstantVelocityFrames..................... 44
2.15 Example:MotioninanInvariantPotentialField ................... 45
2.16 AlternativeEnergy-MassVelocityVariation........................ 49
3 GeneralFormulationandBasicEquations.............................. 55
3.1 Introduction........................................................... 55
3.2 LouisVictordeBroglie.............................................. 56
3.3 JamesClerkMaxwell ................................................ 63
3.4 FourTypesofMatterandVariableRestMass...................... 68
xi