Emergence of complex and spinor wave functions in Scale Relativity. II. Lorentz invariance and bi-spinors

0.17 MB·English

by  Marie-Noëlle Célérier

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1 Emergence of complex and spinor wave
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a functions in Scale Relativity. II. Lorentz
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] invariance and bi-spinors
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e Marie-No¨elle C´el´erier and Laurent Nottale, ∗
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i January 11, 2013
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Abstract
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v
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Owingtothenon-differentiablenatureofthetheoryofScaleRel-
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ativity, the emergence of complex wave functions, then of spinors
2
.
1 andbi-spinorsoccursnaturallyinitsframework. Thewavefunction
0
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is here a manifestation of the velocity field of geodesics of a con-
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:
v tinuous and non-differentiable (therefore fractal) space-time. In a
i
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firstpaper(PaperI),wehavepresentedthegeneralargumentwhich
r
a
leads to this result using an elaborate and more detailed derivation
∗LUTH, Observatoire de Paris, CNRS, Universit´e Paris-Diderot, 5 place Jules
Janssen, 92195 Meudon Cedex, France; e-mails: [email protected], lau-
[email protected],
1
thanpreviouslydisplayed. Wehavethereforebeenabletoshowhow
the complex wave function emerges naturally from the doubling of
thevelocity field and to revisit thederivation of thenon relativistic
Schr¨odingerequation of motion. In thepresent paper(Paper II) we
dealwithrelativisticmotionanddetailthenaturalemergenceofthe
bi-spinors from such first principles of the theory. Moreover, while
Lorentz invariance has been up to now inferred from mathematical
results obtained in stochastic mechanics, wedisplay herea new and
detailedderivationofthewayonecanobtainaLorentzinvariantex-
pressionfortheexpectationvalueoftheproductoftwoindependent
fractalfluctuationfieldsinthesoleframework ofthetheoryofScale
Relativity. These new results allow us to enhance the robustness of
our derivation of the two main equations of motion of relativistic
quantum mechanics (the Klein-Gordon and Dirac equations) which
we revisit hereat length.
KEYWORDS:Scalerelativity;Relativisticquantumtheory;Spinors;
Lorentzinvariance.
1 Introduction
From a physical point of view, the theory of Scale Relativity (SR)
is the generalization to scales of the relativity principle of motion
which underlies the foundation of a large part of classical physics.
Fromamathematicalpointofview,itisthegivingupofspace-time
2
differentiability. Both generalizations result in the fractal nature of
space-time [1].
Oneofthemain achievementsofthistheoryisthefoundationof
quantum mechanics on first principles. In its framework, the quan-
tummechanicalpostulateshavebeenderivedandthecomplex,then
spinorial,thenbi-spinorialnatureofthewavefunctionhasbeennat-
urally recovered [1, 2, 3, 4, 5], while the corresponding quantum
mechanical motion equations, the Schr¨odinger [1], Pauli [3], Klein-
Gordon [6, 7] and Dirac [4, 5] equations havebeen demonstrated.
The theory has also allowed us to generalize the quantum me-
chanicalmotionequationstothemacrocopicrealm. Thisisobtained
whentheratioofthereducedPlanckconstant~overtwicetheelec-
tron mass, operating in the microscopic domain, is given a more
general interpretation in terms of a macroscopic constant whose
D
value is linked to the physical system under study. We have been
thereforeabletoderiveamacroscopicSchr¨odinger-likeequationwith
numerousapplications in physicsand other fields(see, e.g., [8] for a
review) and a macroscopic Dirac-like equation whose non-relativist
limit might tentatively reproducesome turbulentfluid behavior [9].
Theemergenceofthecomplexnumbersandtheirgeneralizations,
thequaternionsandthebi-quaternions,isissuedfromthesuccessive
doublingsofthevelocityfieldsduetothenon-differentiabilityofthe
fractal functions representing the space-(time) coordinates. These
doublingshavebeenextensivelystudiedinpreviousworks(see,e.g.,
3
[8] for a review).
Now, a new and more detailed derivation of the way the two-
valuedness of the mean velocity field emerges naturally in terms of
a (+) and (-) velocity has been put forward in [10] (Paper I). It
yields V and U velocity fields, which are subsequently combined in
termsof thecomplex velocity field =V iU used in thegeodesic
V −
equation d /dt = 0 written in the fractal 3-dimensional space and
V
from whicbh the Schr¨odinger equation is derived. We are led here
to generalize these results to the 4-dimensional space-time, being
therefore allowed to recover the complex 4-velocity field needed to
obtain the Klein-Gordon equation.
Weextendthenthistwo-valuednesstosuccessiveotherdoublings.
We obtain thus naturally a more symmetric expression than that
provided in [4, 5] for the bi-quaternionic velocity field yielding the
bi-spinor and the Dirac equation, from which the spinor and the
Pauli equation has been derived [3].
Another essential tool used to derive the quantum mechanical
motion equations is the expectation value of the product of two in-
dependentfractalfluctuationfields. Itsexpressionappearsmostnat-
urallyinthenon-relativistic3-dimensionalgeneralizationofthethe-
ory,theninits4-dimensionalrelativisticversion. Duetothestochas-
tic nature of these fluctuations, the transition from non-relativistic
to relativistic motion allows us to write a Lorentz invariant expres-
sionforthisexpectationvalueintherelativisticcaseandthereforeto
4
obtain the Klein-Gordon and Dirac equations. The transition from
non-relativisticmotiontotherelativisticKlein-Gordonequationhas
been first implemented in the framework of stochastic mechanics
[11,12,13]andthemathematicaltools thusprovidedhavebeenap-
pliedtothescalerelativisticfractalfluctuationsallowingustoobtain
therelativistic equationsofquantummechanics[6,7,4,5]. Wepro-
pose here a new derivation of the way one can obtain this Lorentz
invariancein thesole framework of SR.
Thestructureofthepresentpaperisasfollows. InSec.2wegen-
eralizetheresultsofPaperItothefullspace-timeoftherelativistic
caseandobtainnaturallytheemergenceofLorentzinvarianceinthe
frameworkofSR,allowingustogiveinSec.3amoreaccuratederiva-
tion of therelativistic Klein-Gordon and Dirac equations. Section 4
is devoted to theconclusions.
2 Lorentz invariance of the relativis-
tic tools of SR
WebeginbyrecallingbelowthemainequationsdisplayedinPaperI
whichwewillneedhereforourpurpose. IthasbeenshowninPaper
IthatthedifferentialelementdX(x(t),t,dt,ε )usedinthedefinition
t
of a fractal velocity field V(x(t),t,dt,ε ) = dX/dt considered on a
t
fractal curve, with the time coordinate t as a parameter along this
5
curve, can be decomposed into a differentiable linear contribution
and a non-linear fractal fluctuation, such as
dX =V[x(t),t]dt+U[x(t),t]ε +dξ. (1)
t
Thefractal fluctuation,dξ,hasbeentheredescribedbyastochastic
variablewhich possesses thenatureof aspaceresolution ε >0,i.e.
x
dξ=ηε , (2)
x
whereη isadimensionless variable,normalizedaccordingto η =0
h i
and η2 =1. Whenthefractaldimension exhibitsthecritical value
h i
D =2 [1, 7, 14], thespace and time resolutions are related by
f
ε =√2 ε . (3)
x t
D
Wehavetherefore obtained
dX =V[x(t),t]dt+U[x(t),t]ε +η√2 ε . (4)
t t
D
On the interface ε = dt, which defines the physical fractal deriva-
t
| |
tive,this becomes
dX =Vdt+U dt +η 2 dt. (5)
| | D| |
p
Two possibilities occur therefore for the elementary displacements:
dt>0: dt =dt, dX+ =(V +U)dt+η√2 dt. (6)
| | D
dt<0: dt = dt, dX− =(V U)dt+η√ 2 dt. (7)
| | − − − D
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By setting v+ =V +U and v− =V U, the two-valuedness of the
−
mean velocity field has been recovered in the non-relativistic case
described in Paper I.
Weconsidernowtherelativisticcaseandderivearelativistically
covariant expression for theexpectation valueof theproduct of two
independentfractalfluctuationfieldsneededtocompletethederiva-
tion of theKlein-Gordon and Dirac equations in SR.
The generalization of the results displayed in Paper I to a 4-
dimensional space-time runs as follows. The curvilinear parameter
along the geodesics is now the invariant s. Not only space, but
thewholecontinuousspace-timeisassumedtobenon-differentiable,
hence fractal. We can write the elementary displacement, dXµ, for
each coordinate xµ, µ=0,1,2,3, along a geodesic of fractal dimen-
sion D =2 as
f
dXµ =Vµ[xµ(s),s]ds+Uµ[xµ(s),s]ǫ +dξµ, (8)
s
with
dξµ=ηµǫxµ, (9)
where the η ’s are dimensionless variables normalized according to
µ
η =0, η η =δ , and
µ µ ν µν
h i h i
ǫxµ =√2 ǫs. (10)
D
At the interface where the derivatives with respect to s take their
physical values whatever the scale, we make the particularly mean-
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ingful choice ǫ = ds which implies
s
| |
dξµ=η √2 ǫ =η 2 ds. (11)
µ s µ
D D| |
p
In thecase ds>0 for each µ, we thusobtain
dξµ =η √2 ds, (12)
+ µ D
and in the case ds<0 for each µ,
dξµ =η √ 2 ds. (13)
− µ
− D
This implies
dξ±µdξ±ν = 2 δµνds. (14)
h i ± D
We consider now that each dξµ behaves such as an independent
fractalfluctuationarounditsownmeanequaltozero,inaccordance
with η = 0. This justifies to consider it as a stochastic process
µ
h i
which we choose to be of Berstein type, since such a type includes
a large class of stochastic processes among which Markov and anti-
Markovprocesses. Weinspireourselvesherefromthereasoningsput
forwardbyGuerraandRuggiero[11]andServa[12]intheframework
of a stochastic diffusion theory, while adapting them to the scale
relativistic concepts.
The reasoning proposed in [11] takes into account the fact that
timeandspaceplayformallyanidenticalroleinrelativistictheories.
The authors of [11] consider therefore that a particle can change
direction not only in space, but also in time. A particle moving
backwardintimecanbeinterpretedasanantiparticleandthepoint
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of inversion of its temporal motion can be interpreted as a point
where pairs of particles are created or annihilated. Generally, the
trajectory of a particle can intersect many times a hyper-surface of
constant time coordinate t. The theory can thus be thought of as
describing a many particle system with creation and destruction of
pairs. Apossibleinterpretationisthatthe4-dimensionalone-particle
process is merely a mathematical “trick” used to describe a many
particle system. The invariant s has therefore no physical meaning.
It is merely a tool used to associate a probability measure to any
possible realization of themany particle process.
Another possible interpretation, put forward in [12] and mathe-
maticallyequivalenttothepreviousone,isthattheone-particletra-
jectories possess a physical meaning and that amicroscopic particle
can actually invert its temporal motion. The invariant parameter s
ishere still a tool used to associate a probability to each realization
butit also appears as a generalization of theproper time.
We want to stress now that, at variance with the method devel-
oped in [11] and [12], we do not consider here the inversion of the
invariantparameters,butthebreakingofthesymmetryds ds.
↔−
However, as regards the above interpretation in terms of backward
motion on the fractal geodesics considered here in a local sense and
creation and annihilation of pairs, an inversion of ds is physically
equivalenttoaninversionoftheparametersitself. Thishasalready
been stressed in [8]. Following a reasoning inspired by Serva [12],
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we consider two classes of stochastic processes that both contain a
combinationofMarkovianandanti-Markovianprocessesandcanbe
thoughtof as Berstein processes: thefirst with a positiveincrement
of the s parameter for the time coordinate and negative increments
of this s parameter for the space coordinates, the second with a
negative increment of the s parameter for the time coordinate and
positive incrementsof this s parameter for thespace coordinates.
We choose thus, for the first process denoted 1, ds > 0 for the
X0 coordinate, which gives
1
dX0 =V0ds+U0ds+η0√2 ds, (15)
1 1 1 1
D
which we write
dX−0 =v−0ds+η−0√2 ds=v−0ds+dξ−0, (16)
D
with v−0 = V10+U10. And we choose conversely, for this process 1,
ds<0 for theXi coordinates, which yields
1
dXi =Vids Uids+ηi√ 2 ds, (17)
1 1 1 1
− − D
which we write
dX−i =v−i ds+η−i √ 2 ds=v−i ds+dξ−i , (18)
− D
with v−i =V1i U1i. The upper index 0 denotes here the time coor-
−
dinate, while i=1,2,3 denotes thespatial coordinates.
For the second process denoted 2, we choose ds < 0 for the X0
2
coordinate, which gives
dX0 =V0ds U0ds+η0√ 2 ds, (19)
2 2 2 2
− − D
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