A consistent unified framework for the new system of units: Matter–wave optics

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A consistent unified framework for the new system of units: Matter–wave optics

Christian Bordé

To cite this version:

Christian Bordé. A consistent unified framework for the new system of units: Matter–wave optics. Comptes Rendus. Physique, Académie des sciences (Paris), 2019, 20 (1-2), pp.22-32.

�10.1016/j.crhy.2018.12.004�. �hal-02164995�

Contents lists available atScienceDirect

Comptes Rendus Physique

www.sciencedirect.com

The new International System of Units / Le nouveau Système international d’unités

A consistent unified framework for the new system of units:

Matter–wave optics

Un cadre unifié cohérent pour le nouveau système d’unités : l’optique des ondes de matière

Christian J. Bordé

^a

^,

^b

^{, ∗ ,}

¹

aLNE-SYRTE,UMR8630CNRS,ObservatoiredeParis,61,avenuedel’Observatoire,75014Paris,France

bLaboratoiredephysiquedeslasers,UMR7538CNRS,UniversitéParis-Nord,99,avenueJean-Baptiste-Clément,93430Villetaneuse,France

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Availableonline11January2019

Keywords:

SIunits Metrology Geometry Action Entropy Matter–waves

Mots-clés : UnitésSI Métrologie Géométrie Action Entropie Ondesdematière

Thankstoconsiderableprogressinquantumtechniques,ithasbeenpossibletoredefineall SIbaseunitsfromfundamentalconstants.Alastchallengeistoproduceaunifiedframe- work forfundamental metrologyin whichallbase quantitiesand relevant fundamental constantsappear naturallyand consistently.We suggest ageneralized 5Dframework in whichboth gravito-inertialand electromagneticinteractions have a naturalgeometrical significationandinwhichallmeasurements canbereducedtophasedeterminationsby opticalormatter–waveinterferometry.Thecornerstonesofthisunificationareactionand entropy.TheconnectionismadewithKaluza’s5DtheoryandPlanck’snaturalunits.

©^{2018Académiedessciences.PublishedbyElsevierMassonSAS.Thisisanopenaccess} articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

r é s u m é

L’année2018 aété celledu renouvellement complet du systèmed’unités, avec l’ambi- tiond’établir un systèmepérenne, universel et cohérent. Ce systèmese caractérise par l’abandondesartefactspoursefonderuniquementsurlesconstantesfondamentalesdela Physique[1–3]. Lacohérencesous-jacente dusystèmeproposéetlanotiondeconstante fondamentaleseront discutées.Un système d’unitésnaturelles aété introduit par Max Plancken1899.Ilestfondé surcinqconstantesfondamentales,^h,¯ ^kB,c,G,

ε

0.Lesdeux premièresconcernentrespectivementlesmouvementscohérentsetladécohérencether- miquedans l’espacedes phasesaumoyendes conceptsd’actionetd’entropie. Lestrois dernièresprécisentlagéométriedecetespaceenprésenced’interactionsgravitationnelles etélectromagnétiques.Cettegéométrieestcelled’unespaceàcinq dimensionsintroduit parTheodorKaluzaen1921[4].Lesprogrèsconsidérablesdesinterféromètresàondesde matièredanslesdomainesatomiqueetélectriqueimposentaujourd’huiunnouveausys- tème,danslequellesdeuxdernièresconstantessontplutôtunedifférencedemasseentre lesniveauxd’unmêmeatomeetlachargedel’électron.Lacinquièmedimensionestalors letempspropredonnéparleshorlogesatomiques.Lelienentrelesdeuxsystèmesfaitin-

*

Correspondenceto:LNE-SYRTE,UMR8630CNRS,ObservatoiredeParis,61,avenuedel’Observatoire,75014Paris,France.

E-mailaddress:[email protected].

1 Memberofthe“InstitutdeFrance”,AcadémiedesSciences,Presidentofthe“Conférencegénéraledespoidsetmesures”(CGPM)in1999,2003,2007, and2011.

https://doi.org/10.1016/j.crhy.2018.12.004

1631-0705/©^{2018Académiedessciences.PublishedbyElsevierMassonSAS.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense} (http://creativecommons.org/licenses/by-nc-nd/4.0/).

tervenirlaconstantedestructurefineetsonéquivalentgravitationnel.Lesystèmeproposé apparaîtalorscommelemeilleurcompromis,dansl’étatactueldenosconnaissances.

©^{2018Académiedessciences.PublishedbyElsevierMassonSAS.Thisisanopenaccess} articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction:whatframeworkforfundamentalmetrology?

Thisframeworkisnaturallytheoneimposedbythetwogreatphysicaltheoriesofthe20thcentury:relativityandquan- tummechanics.Thesetwomajortheoriesthemselveshavegivenbirthtoquantumfieldtheory,whichincorporatesalltheir essentialaspectsandaddsthoseassociatedwithquantumstatistics.Thequantumtheoryoffieldsallowsaunifiedtreatment offundamentalinteractions,especially,ofelectroweakandstronginteractionswithinthestandardmodel.GeneralRelativity isaclassicaltheory;hence,gravitationremainsapartandisreintegratedintothequantumworldonlyintherecenttheo- riesofstrings.Wedonotwishtogothatfarandwewillkeeptoquantumelectrodynamicsandtotheclassicalgravitation field. Such a framework is sufficient to build a modern metrology, takinginto account an emerging quantummetrology [1–3]. Ofcourse, quantumphysics has beenoperating fora long time attheatomic level,forexample inatomic clocks, butnowitalsofillsthegapbetweenthisatomicworldandthemacroscopicworld,thankstothephenomena ofquantum interferences,whetherconcerningphotons,electrons,Cooperpairsor,morerecently,atomsinatominterferometers[5–10].

The mainconsequence ofthisevolution is thatall base unitscan nowbe redefined byfixing the valuesof fundamental constantshavingadimension,suchasc,h,kB...whereasconstantswithoutdimensionsuchasthefinestructureconstant

α

cannotbearbitrarilyfixed.²

1.1. Fundamentalconstantsandsymmetries

Somequantitiestransformintoeachotherinsymmetrytransformations,thusreflectingacommonnature.Ifindependent unitshavebeenchosenforthecorrespondingquantities,afundamentalconstantappearsasaresultofthisarbitrarychoice.

Thisisthecaseforspace–timecoordinatesconnectedtogetherbyrotationsorbyLorentztransformationsorelsetoproper time as a result ofgauge transformations. Other pairs of quantities can be shown to be equivalent once their common natureisunderstoodandrecognized.Thisisthecaseofenergyandthermalexcitation.Thisequivalenceisresponsiblefor theBoltzmannconstant.Similarly,massisequivalenttoaproperfrequencythroughPlanck’sconstant.Inallthesecases,the resultingconstantmaybefixed(andeveninsomecasessetequaltounity).Finallynatureimposessomesystemsforwhich aquantityisquantizedhencebecominguniversalandreproducibleforidenticalobjects.Thisisthecaseofatomiclevelsor ofthemassofelementaryparticles,thusimposingaunitoftime.

1.2. Geometryandnumbers

Mostbasequantitiesofmetrology,length,time,mass,electricalquantities,temperature,areultimatelymeasuredbyop- ticalormatter–waveinterferometers.Opticsandquantummechanicsplayacentralroleinthedescriptionofthesedevices.

Asaconsequence,futurefundamentalmetrologywilldealessentiallywithphasemeasurementsi.e.invariantnumbers.One shouldalsoemphasizethenon-commutingcharacterofquantitieslikemassandpropertime,whichisareasonwhyPlanck’s constanthassuch aspecialplaceinthesystemofunits.Basequantitiesshouldbe quantumobservables.Someappearas basequantitieswiththeir conjugatepartner (e.g.,massandpropertime),othersdonot(e.g.,positioncoordinateandmo- mentum).Thequantum-mechanicallinkbetweenconjugatequantitiesdoesnotallowanymoretoleave Planck’sconstant outofthesystemofunits.Quantummeasurementtheorybecomesessentialtoexplorethelimitsofthenewmetrology.

Anatural5Dtheoreticalframework fortheredefinitionoftheSIispresented,inwhichaclearseparationismadebe- tweenpropertime(observable!)andtimecoordinate(notobservable!)asdistinctquantitiessharingthesameunit.Therole ofthe electromagneticfield isto couplespace–timeandproper time coordinatesthrough thecorresponding off-diagonal componentsofthemetrictensor.The5Dactiongathersallphenomenaandconstantsofinterestforafullyrelativisticquan- tummetrologyinaninvariantphasethroughPlanck’sconstantandthisincludesthedephasingarisingfromgravito-inertial fields(e.g.,the Sagnaceffect ortheeffect ofgravitationalwaves) aswell asthoseofelectromagnetic origin(suchasthe Aharonov–BohmortheAharonov–Cashereffect).

1.3. Actionandentropy

Owingto itslinkwiththequantummechanicalphase throughPlanck’sconstant,theconcept ofactionthereforeplays thecentralroleinthenewmetrology.MassisdirectlyconnectedtoPlanck’sconstant,notonlyinatomicclocksphasesbut

2 Theseconstantsareratiosofquantitieswithidenticalunitsandtheyhavetobemeasuredandinsomecasescanbecalculated.

Fig. 1.CoherentandincoherentmotionsleadtoadimensionalfactorsthatmaybeexpressedandclassifiedwithPlanckandBoltzmannconstantsalongthe sidesofthistriangle.

alsointhe recoilshift[11,12,16],nowaccessiblewithhighaccuracybyatom interferometry[14,15].³ Ournewsystemof unitscannotescapethefactthatmassandtimeunitsaredirectlycoupledthroughthenaturalstandardofactionwhichis Planck’sconstant.Thislinkisfurtherreinforcedbythepossibilitytodaytoextendthisconnectiontothemacroscopicworld thankstosiliconspheres[19,20] andtothewattbalance[21].

Statistical mechanicspermitstogofromprobabilitiestoentropythanks toanotherdimensionedfundamentalconstant, Boltzmann’s constant kB.By analogywiththe caseof Planck’sconstant, it seems naturalto fixBoltzmann’s constant kB. Indeed,thereisadeepanalogybetweenthetwo“S’s”ofphysics,whichareactionandentropy(illustratedinFig.1).They providerespectivelythephasesandtheamplitudesforthedensityoperator[22]:

ρ =

ψ

p

_ψ

|ψ ψ|

ThecorrespondingconjugatevariablesofenergyarerespectivelytimefortheLiouville–vonNeumannequationandrecipro- caltemperatureintheBlochequation.

The link betweenatom interferometry andtheDoppler broadeningof lineshapesby thethermal motionofatomsis established in reference [13] which brings the connection between phase and temperature measurements. The thermal motionofatomsisresponsibleforalossofphasecoherenceandtheDopplerbroadeningmaybeseenasalimitedvisibility ofinterferencefringes.TherelativeDopplerwidthisindeedameasureofentropy⁴andisoneofthemethodsthathaveled toasatisfactorydeterminationoftheBoltzmannconstant[23].

Inordertocomputeactionandentropy,weneedtoknowthefull5Dgeometryandthenfromtheexpressionsofaction andentropy,wecanobtainthedensityoperatorandhencetheaveragevalueofanyquantity.

2. From4to5dimensions:propertimeintervalandmassasaconjugatepairfora5-dimensionaloptics

Matter–waveinterferometrystartedwithelectronsandneutrons,forwhichapurespace–timedescriptionissufficientin analogywithordinaryinterferometryoflightwaves.However,anextraphasefactorisrequiredforthesemassiveparticles, whichisprovidedbytheaction.Since,fortheseparticles,massisaconstantofmotion,thecorrespondingphaseissimply proportional tothepropertime intervalalongeachpath.Whenitcomestoatomsormolecules,massvarieswithinternal excitationanditplays thesameroleasanyexternalmomentumcomponentasanewdynamicalvariabletoachievemode couplingandgeneratenewopticalpathsinaninterferometer.Theinternalphasecontributestotheoveralldephasingand isresponsiblefortheclockterminatoms.Atomicclockscanthenbeseenasgenuineatominterferometers.Toaccountfor thisinternalmotion,wemayenlargeourspace–timewiththeadditionaldimensions

=

^c

τ

(seeFig.2)

d x

^μ

= ( c dt , dx , dy , dz , c d τ ) = ( dx

⁰

, dx

¹

, dx

²

, dx

³

, dx

⁴

) (1)

3 TheratioofPlanck’sconstanttoelectronmasswasalreadymeasuredwith muchloweraccuracyusinginterferometryinrotatingsuperconductors [17,18].

4 AninterestinganalogymaybedrawnforthetwoinaccessiblelimitsthatarethevelocityoflightcandtheabsolutezerotemperatureT=0.Inboth cases,internalmotionstopsandbothvelocitiesdτ/dt(cf.Langevintwins)andu=

2kBT/m−→^{0.(TheDopplerwidthandtheblackbodyradiationshift} vanishasthethermaldecoherencetimedecreases.)

Fig. 2.In5Dcoordinates,the5Dvelocityisalwaysc,buttheprojectiononspaceistheparticle’svelocity−→_{v,whereasthecomponentalongtheproper} timeaxisisreducedaccordingly.

andintroduceageneralizedlightconeformassiveparticles

d σ

²

₌ G

_μˆνˆ

d x

^μˆ

d x

^ν

= ^{0 with} μ ˆ , ν ˆ ₌ 0 , 1 , 2 , 3 , 4 (2)

whereGμν

=

^gμν;Gμˆ4

=

^G4νˆ

=

^0;G44

=

^G44

= −

^1.

Inflatspace–time,since g⁰⁰

=

^{1 andgii}

= −

^{1,thisrelationreducesto}Pythagoras’theorem:

c

²

dt

²

= c

²

d τ

²

+ d − → _x

2

(3)

Propertimeisnotdefinedbythisequationfromothercoordinates,butisa trueevolution parameterrepresentativeof theinternalevolutionoftheobject.Itcoincidesnumericallywiththetimecoordinateintheframeoftheobject.Sincethe timeunitisapropertimeunitprovidednowbyanatomicclock,thischoicedeterminesthetimecoordinatescaleandthe lengthunitbyfixingthevelocityoflight.

Massisconservedwhen thesystemunderconsideration isinvariantinaproper timetranslationandwillbecomethe generator ofsuch translations inthe quantum theory.In thecaseof atoms, the internal degreesoffreedom give rise to a massthat varieswiththe internalexcitation. Forexample,in thepresence ofan electromagneticfield inducing transi- tions betweeninternal energylevels,the massofatoms becomes time-dependent (Rabioscillations).It isthus necessary toenlargetheusualframeworkofdynamicstointroducethisnewdynamicalvariableasafifthcomponentoftheenergy–

momentumvector.

Theenergy–momentumrelation

E

²

= − → _p

2

c

²

+ ^m

²

^c

⁴

⁽⁴⁾

canbewrittenwithafivedimensionalnotation:

G

^μˆνˆ

p

_μˆ

p

_νˆ

= ^{0 (5)}

where

^pμ_ˆ

= (

pμ

,

p₄

= −

^mc

)

(seeFig.3).

Finally,ifwecombinemomentaandcoordinatestoformamixedscalarproduct,weobtainanewrelativisticinvariant, whichisthedifferentialoftheaction.In5D,weshallthereforeintroducethesuperaction:

S = −

p

_μˆ

d x

^μˆ

(6)

equivalentto

^p

μ_ˆ

= − ∂ S

∂ ^x

^μˆ

^with μ ˆ = ⁰ , 1 , 2 , 3 , 4 (7)

Ifthisissubstitutedin

G

^μˆνˆ

p

_μˆ

p

_μˆ

= ^{0 (8)}

weobtaintheHamilton–Jacobiequationin5D

G

^μˆνˆ

∂

_μˆ

S ∂

_νˆ

S = ^{0 (9)}

Fig. 3.5D energy–momentum picture.

which hasthesameformastheeikonal equationforlight in4D. Itisalreadythisstrikinganalogy thatpushed Louisde Broglietoidentifyactionandthephaseofamatterwaveinthe4Dcase[24].Weshallfollowthesametrackforaquantum approachinour5Dcase.

What is the link between the three previous invariantsgiven above? As in optics, the direction of propagation ofa particleisdetermined bythemomentumvector tangenttothetrajectory.The 5Dmomentumcan thereforebe writtenin theform:

p

^μˆ

= ^d x

^μˆ

/ d λ (10)

where

λ

^{isanaffineparametervaryingalongtheray.Thisisconsistentwiththeinvarianceofthesequantitiesforuniform} motion.

In4Dthecanonical4-momentumis:

p

_μ

= mc g

_μν

dx

^ν

g

_μν

dx

^μ

dx

^ν

= m c g

_μν

u

^ν

(11)

whereuν

=

^dxν

/

cd

τ

isthenormalized4-velocitywithcd

τ =

gμνdxμdxν. Weobservethatd

λ

canalwaysbewrittenastheratioofatimetoamass:

d λ = ^d τ

m = ^dt m

^∗

= ^d θ

M = ... (12)

where

τ

is the proper time of individual particles (e.g., atoms in a clock or in a molecule), t is the time coordinate of the composed object (clock,interferometer, or molecule)and

θ

its proper time;m

,

m^∗

,

M are, respectively, the mass, the relativistic mass of individual particles and their contribution to the scalar mass of the device or composed ob- ject.

Intheusual paradigmofrelativity,thetime t isacoordinatevariableandthepropertime

τ

istakenastheevolution parameter todescribethemotionofparticlesinspace–time.Inthispresentationhowever,proper timeisan independent coordinate describingthe internalmotionof massiveparticles,so thatwe shallrather choosethe coordinatetime asthe evolutionparameter.Weshallthereforewritein5D:

p

_μˆ

= ^m

^∗

^G

μ_ˆνˆ

˙ x

^ν

= ^m

^∗

˙ x

_μˆ

(13)

expressedwiththe“relativisticmass“:

m

^∗

= ^{m dt} d τ ⁼

m c

g

_μν

x ˙

^μ

x ˙

^ν

(14)

andwherethedotreferstoderivationwithrespecttoa“laboratorytime”(identicaltothepropertime

θ

oftheapparatus onlyintheabsenceofgravitationorinertialeffects).Withthischoice

^x

˙

⁰

=

^{c and}

^p0

=

^{m∗c.Analternatechoicecouldbeto} takethepropertime

θ

ofthefulldeviceastheevolutionparameter.Inwhichcase:

c d θ =

G

₀₀

dx

⁰

and M = ^m

^∗

G

₀₀

(15)

From

d σ

²

₌ G

_μˆνˆ

d x

^μˆ

d x

^ν

= ^{0 (16)}

weinferin5D

d S = ^{0 (17)}

Weshallgeneralizetheserelationstoanobject,suchasaclock,amolecule

. . .

,composedofanumberofsubparticlesand illustratethe originofpropertime ascoming fromthe innerstructure oftheobject.Forsuch compositeobjects,massis givenbytheeigenvaluesoftheirinternalHamiltonianandthusbecomesanoperatorinthequantumtheory.

This picture can be generalized withthe introduction of the electromagnetic interaction potentialsin the 5D metric tensor.

3. Generalizationinthepresenceofgravitationalandelectromagneticinteractions 3.1. Generalizationofthemetrictensorfromthegeneralizedinterval

The previous 5D scheme can be extended to generalrelativity with a 4D metric tensor gμν andan electromagnetic 4-potential Aμ

g

^μν

p

_μ

− ^{q A}

μ

( p

_ν

− ^{q A}

ν

) = ^m

²

^c

²

⁽¹⁸⁾

(q

= −

^{efortheelectron).}

WeshallsearchforametrictensorGμ_ˆνˆ for5Dsuchthatthegeneralizedintervalgivenby:

d σ

²

₌ G

_μˆνˆ

d x

^μˆ

d x

^ν isaninvariant.

Letusrecallthat,fromtheequivalenceprinciple,themetrictensor gμν canbeobtainedfromtheMinkovskiflatspace–

time tensor

η

^μν usinginfinitesimal frame transformationsfroma locallyinertialframe. Quitegenerally, anyinfinitesimal coordinatetransformationconsideredasagaugetransformationcanbeusedtointroduceacomponentofthegravito-inertial field.Asanexample,in4D,thetransformation(caseofarotation):

dx

ⁱ

= ^dx

ⁱ

+ α

₀ⁱ

dx

⁰

dx

⁰

= dx

⁰

(19)

transformstheinterval

ds

²

= ^g

00

( dx

⁰

)

²

+ ^g

_{i j}

^dx

ⁱ

^dx

^j

⁽²⁰⁾

into

ds

²

= ^g

00

( dx

⁰

)

²

+ ^2g

0i

dx

⁰

dx

ⁱ

+ ^g

i j

dx

ⁱ

dx

^j

(21)

with

g

00

= ^g

00

+ α

₀ⁱ

α

₀^j

g

_{i j}

(22)

g

0j

= α

₀ⁱ

g

_{i j}

(23)

g

i j

= g

_{i j}

(24)

g

⁰⁰

= ^g

⁰⁰

= ¹ / g

₀₀

(25)

g

^{i j}

= ¹ / g

_{i j}

(26)

Using

g

i j

g

ⁱ⁰

= − ^g

⁰⁰

^g

j0

(27)

wefind

α

ⁱ₀

_{= −} ^g

i0

g

⁰⁰

(28)

α

ⁱ₀

α

₀^j

g

_{i j}

= − ^g

ⁱ⁰

^g

ⁱ⁰

g

⁰⁰

(29)

Inthecaseofrotationwerecovertheusualmetrictensorintherotatingframe.

Theaction Sbecomes

S = −

p

_μ

dx

^μ

= −

p

₀

dx

⁰

−

p

_i

dx

ⁱ

(30)

= −

p

₀

dx

⁰

−

p

_i

( dx

ⁱ

+ α

ⁱ₀

dx

⁰

) (31)

S = − ^p

₀

+ ^p

_i

α

₀ⁱ

dx

⁰

−

p

_i

dx

ⁱ

= −

p

_μ

dx

^μ

whichgivestheSagnacphaseas p_igⁱ⁰

/

g⁰⁰

dx⁰.

Thesameapproachcanbeusedwiththefifthdimensionbyintroducingthegaugetransformation

dx

⁴

= ^dx

⁴

+ β

_μ⁴

d ^x

^μ

d ^x

^μ

= ^{d x}

^μ

⁽³²⁾

togeneratetheoff-diagonalelementsGμ4

d σ

²

= ^G

44

dx

^{4 2}

+ ^{2 G}

44

β

_μ⁴

dx

⁴

d ^x

^μ

+

g

_μν

+ β

_μ⁴

β

_ν⁴

G

₄₄

d ^x

^μ

^{d x}

^ν

G

₄₄

= ^G

₄₄

⁽³³⁾

G

_μ4

= β

_μ⁴

G

₄₄

(34)

G

_μν

= ^g

μν

+ β

_μ⁴

β

_ν⁴

G

₄₄

(35)

Thesuperaction

S givenby(6) becomes

S = −

p

_μˆ

d x

^ˆμ

= −

p

_μ

d x

^μ

−

p

4

dx

⁴

(36)

= −

p

_μ

d x

^μ

+

m c ( dx

⁴

+ β

_μ⁴

d x

^μ

) (37)

S = − ^p

μ

− ^{m c} β

_μ⁴

d ^x

^μ

+

m c

²

d τ (38)

whichyieldstheAharonov–Bohmphaseifmc

β

_μ⁴

=

^qAμ.

ThemetrictensorinfivedimensionsGμν isthuswrittenasinKaluza’stheorytoincludetheelectromagneticgaugefield potential Aμ

G

_μˆνˆ

=

G

_μν

G

_μ4

G

4ν

G

44

=

g

_μν

+ κ

²

G

₄₄

A

_μ

A

_ν

κ G

₄₄

A

_μ

κ G

₄₄

A

_ν

G

₄₄

G

^μˆνˆ

=

G

^μν

G

^μ4

G

^4ν

G

⁴⁴

=

g

^μν

− κ A

^μ

− κ A

^ν

G

⁴⁴

(39)

where

κ

isgivenbythecharge-to-massratiooftheobject.Thismetrictensorissuchthat

G

^μˆλ

G

_λνˆ

=

G

^μλ

G

^μ4

G

^4λ

G

⁴⁴

G

λν

G

λ4

G

_4ν

G

₄₄

= δ

^μ_νˆ^ˆ

(40)

=

G

^μλ

− κ A

^μ

− κ A

^λ

G

⁴⁴

g

λν

+ κ

²

G

₄₄

A

λ

A

_ν

+ κ G

₄₄

A

λ

+ κ G

44

A

_ν

G

44

=

G

^μλ

g

λν

κ G

₄₄

G

^μλ

A

λ

− κ G

₄₄

A

^μ

= ⁰

− κ A

^λ

( g

_λν

+ κ

²

G

₄₄

A

_λ

A

_ν

) + κ G

₄₄

G

⁴⁴

A

_ν

− κ

²

G

₄₄

A

^λ

A

_λ

+ ^G

⁴⁴

^G

44

= δ

_ν^μ_ˆ^ˆ

whichimplies

G

^μλ

g

λν

= δ

^μ_ν

G

⁴⁴

= ¹ / G

₄₄

+ κ

²

A

^λ

A

λ

(41)

Theequation

G

^μˆνˆ

p

_μˆ

p

_νˆ

= ^{0 (42)}

with

p

_μˆ

= ( p

_μ

, − ^mc ) (43)

andG44

= −

^{1 isthereforeequivalenttoequation(18)}

g

^μν

p

_μ

− ^{q A}

μ

( p

_ν

− ^{q A}

ν

) = ^m

²

^c

²

⁽⁴⁴⁾

Higher-order electromagneticinteractions are introduced via the multipolar expansion pμ

−

^{q A}μ

+

^QλFμλ, wheredipole momentswillbecomeoperatorsinthequantumdescription.⁵

Thefundamentalconstantsassociatedwiththisgeometryanditsmetrictensorare:

c , m, and e (45)

whichtogetherwith^h

¯

^{andkB constitutethenaturalsetof}fundamentalconstants tobe fixedforthenewsystemofunits.

Thechoiceofthereferencemassmismadeaccordingtothebestpossibleclockrequiredtodefinetheunitofpropertime.

Clearly,todaythismasshastobethemassdifferenceoftheatomiclevelscorrespondingtothetransitiondefiningthisunit.

Wecanderivethedirectlinkbetweenthecoordinate

x⁴ andthepropertime

τ

startingfrom:

G

_μˆνˆ

d x

^μˆ

d x

^νˆ

= ⁰

weobtain

dx

⁴

= − ^G

^4μ

G

44

d x

^μ

+ ¹ G

44

G

4μ

G

4ν

− G

44

G

_μν

1/2

d x

^μ

d x

^ν

1/2

= − ^G

^4μ

G

₄₄

d x

^μ

+ √ ¹

− ^G

44

g

_μν

d x

^μ

d x

^ν

1/2

which,comparedto

dx

4

= ^G

44

dx

⁴

+ ^G

4μ

d x

^μ gives

dx

4

= −

− ^G

44

g

_μν

d x

^μ

d x

^ν

1/2

= −

g

_μν

d x

^μ

d x

^ν

1/2 3.2. Complexbodies(compositeobjects)

Wecanconnectthispropertimewithanintra-atomicorintra-molecularmotionandthuslinkmassandinternalkinetic energy.Ifwe consideran object(suchasa clock,amolecule...)composedofanumberofsubparticles,the5Dsuperaction differentialisgivenbythesum:

d S =

A

− ^p

Aμ

dx

^μ_A

+ ^m

A

c

²

d τ

_A

(46)

wherem_AisthemassofparticleA.Withthefollowingchangeofcoordinates:

dx

^μ_A

= ^dX

^μ

+ ^d ξ

_A^μ

(47)

d S = − ^P

μ

dX

^μ

+

A

− ^p

Aμ

d ξ

_A^μ

+ ^m

A

c

²

d τ

A

(48)

with

P

_μ

=

A

p

_Aμ

and p

_Aμ

= m

^∗_A

/ m

^∗

P

_μ

+ π

Aμ

(49)

Thecoordinates Xμand

ξ

_A^μaresuchthat

A

m

^∗_A

d ξ

_A^μ

= ^{0 and} ξ

_A⁰

= ^{0 (50)}

(commontimecoordinateforalltheparticlesofthecomposedobject).Oneobtainsforthefullobject:

d S = − ^P

μ

dX

^μ

+ ^Mc

²

^d θ = − ^P

μ_ˆ

dX

^μˆ

= ^{0 (51)}

5 Foratomsinteractingwithlight,thesedipolemomentoperatorsareoff-diagonalwithrespecttointernalstates,i.e.massstatesandequation(42)leads tocoupledequationsforthedifferentstates.Asexamples,references[28,10] presentcoupledDiracequationsfortwo-levelsystems.

providedthat:

Mc

²

d θ =

A

− ^p

Aμ

d ξ

_A^μ

+ ^m

A

c

²

d τ

A

(52)

=

A

−π

A j

d ξ

_A^j

+ ^m

A

c

²

d τ

A

(53)

This calculation canbe easily generalizedto take into accountthe electromagneticinteraction potentialsbetweenthe constitutiveparticles:

A

− ^p

Aμ

q

_A

m

A

A

^μ

d x

A4

(54)

In all cases, the source of the proper time

θ

for the object lies in the internal degrees of freedom and its mass Mc² is given by its internal Hamiltonian. A well-defined quantum phase for the composed object requires that it should be inan eigenstateofthisinternal Hamiltonian.Forobjectslikeatoms,molecules...,massisthusan operatorwithquantized eigenvalues.Thedescriptionoftheinteractionwithanexternalelectromagneticfieldrequirestodealwithcoupledequations corresponding toeachofthesemasseigenstates.Awaveequation willbewrittenforeachinternalstateandeachofthese iscoupledwiththeotheronesthroughanoff-diagonalelectricdipoleoperator.

3.3. ConnectionwithKaluza’stheory

In his search for a unified theory including both gravitational and electromagnetic interactions, Theodor Kaluza has alreadyintroduceda5D space–timein1921[4].HisfifthdimensionwaslaterinterpretedbyO.Kleinasaninternalclosed dimensionofelectrons.Weshallnowestablishthelinkbetweenthisadditionaldimensionandthepropertimeusedinour approach.

TheintervalsquaredinKaluza’stheoryiswrittenas:

d σ

²

= ^g

μν

dx

^μ

dx

^ν

+ φ

²

k A

_ν

dx

^ν

+ ^dx

⁵

2 correspondingtothemetrictensor:

G

_μν

=

G

_μν

G

_μ5

G

5ν

G

55

=

g

_μν

+ ^k

²

φ

²

A

_μ

A

_ν

k φ

²

A

_μ

k φ

²

A

_ν

φ

²

G

^μν

=

G

^μν

G

^μ5

G

^5ν

G

⁵⁵

=

g

^μν

− k A

^μ

− ^{k A}

^ν

^k

²

^g

μν

A

^μ

A

^ν

+

_φ¹2

(55)

where

φ

isascalardilatonfieldandwhere

k = ⁴

√ π ^G ε

0

c

Weshifttopropertimeasthenewvariable:

G

44

dx

⁴

= i dx

⁴

= φ dx

⁵ Kaluza’sintervalbecomes

d σ

²

₌ g

_μν

dx

^μ

dx

^ν

+ φ

²

k A

_ν

dx

^ν

+

√ G

₄₄

φ ^dx

4

2

tobeidentifiedwithour5Dinterval:

G

_μˆνˆ

d x

^μˆ

d x

^νˆ

=

g

_μν

+ κ

²

G

44

A

_μ

A

_ν

d x

^μ

d x

^ν

+ ² κ G

44

A

_ν

d x

^ν

dx

⁴

+ ^G

44

dx

^{4 2}

Then:

φ = G

44

κ

i k = i κ

k

isgivenbytheratioofcharge-to-massratiosoftheelectronandofPlanck’sparticle,and

dx

⁴

dx

⁵

= − ⁱ φ = κ

k = ^e m c

c 4 √

π G ε

0

= ¹ 2

α

α

G

Fig. 4.LinkbetweenPlanck’sunitsandproposedSIunitsthankstothefinestructureandtothegravitationalconstants.Thepresentdevelopmentofatomic clocksandofquantumelectricmetrologyobligesustobackupfromPlanck’sunitstoamorerealisticchoicefortimeandelectricalunits.Similarly,the electronmassisnotanoptionfortodayandshouldbereplacedbythemassdifferencebetweentheinternalenergylevelsofanatom.Thisconnection involvesagainadimensionlessconstantsuchasthefinestructureconstantfortheRydbergconstantinthecaseofhydrogentransitions.

Thedispersionrelationsarerespectively:

g

^μν

p

_μ

p

_ν

− 2k A

^μ

p

_μ

p

5

+

k

²

g

_μν

A

^μ

A

^ν

+ ¹ φ

²

( p

5

)

²

= 0

and

g

^μν

p

_μ

− ^{q A}

μ

( p

_ν

− ^{q A}

ν

) = ^m

²

^c

²

⁽⁵⁶⁾

whichgivesthemomentum:

p

5

= − i φ m c = e / k

connectedtochargeasexpected.

ThemetrictensorintroducedbyKaluzaimposesasetofnaturalunitswhichcoincideswiththoseintroducedbyPlanck in1899.Inthissystemthepropertiesofelectrons aregivenbytwodimensionlessconstants, whicharethefinestructure constant andits analogfor gravitation.Atthe presenttime,Planck’s systemofunitsis toofar fromarealistic systemof units,essentiallybecause G cannot bemeasured withsufficient accuracyandcannotcompete withpresentatomicclocks (seeFig.4). Our5Dgeometryusingpropertimeismoregeneral,sinceitappliestoanycompositeobject,includingofcourse atomsandmoleculesthat areusedasclocks.Themassofelectronsappearsthenasthemostelegant choicetodefinethe unitofpropertime.Unfortunately,wedonotknowyethowtobuildagoodclockbaseduponelectron–positronannihilation, andanatomictransitionisthebestcompromisecombininghighaccuracyandacloseconnectionwiththeelectron’smass.

The future choice of the time unit is a fascinating question that is widely debated today and evolves quicklywith the permanentprogressofclocksin theoptical domainandbeyond[25–27]. Finally,giventherole ofgyromagneticratiosin thetheory,consistencyimposestheelectronchargetodefinetheunitofelectricchargeratherthantheelectricproperties ofvacuum,whichisthecasetoday.

3.4. Conclusions

Themostbasicmathematicaltoolofstatisticalquantummechanics,whichisthedensityoperator

ρ

,involvesbothaction andentropyandhencerequiresbothPlanck’sconstant

¯

^handBoltzmann’sconstantkB.Thisoperatoractsinaphasespace that, inturn, contains thegeometry andthe threecorresponding fundamental constants c,

m_at

,

e thanks to themetric tensor.

Action andentropyare thetwocorner stonesofthefutureSI.Theyshouldbe explicitlyintroduced initsformulation.

Massandtime unitsarecoupledthrough^h

¯

^{andbothfixedbythechoiceofthepairofatomiclevelshavingthemassdif-} ference

m_at correspondingtothetransitionBohrfrequency

m_atc²

/

h.Theratio

m_at

/

m_e isobtainedwithhighaccuracy fromtheRydbergconstant.

A naturaltheoretical framework for the redefinition ofthe SI iscompletely provided by the connection between 5D geometry,metrictensor,andmetrologythatwehaveoutlined.Toreducethetheoryofmeasurementstothedetermination of quantum phaseswas our primary objective. The direct link betweenaction and matter–wave interferometry can be foundinanumberofreferencesdealingwithallrecentapplicationsofthisfieldtometrology.Thisapproachisanattempt tounifyallaspectsofmodernquantummetrology.Thefinal aimis,ofcourse,toadopta systemofunitsfree ofarbitrary andartificialfeatures,inharmonywithcontemporaryphysics.Theperspectivethatwehaveadoptedincorporatesnaturally allrelevantfundamentalconstantsinalogicalscheme,withobviousconstraintsofeconomy,aesthetics,andrigour.

Acknowledgement

TheauthorisespeciallygratefultoDr.ChristopheSalomonforveryconstructivecommentsaboutthispaper.

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