FERMIONSANDBOSONS:DETERMINANTSANDPERMANENTS

FROMPHYSICSTOTHETHEORYOFRANDOMPOINTPROCESSES

February 6,2019Lille

OdileMacchi,Membredel’Académiedessciences

OUTLINE

1.Thecontext ofresearch inphysics aroundmy thesis

2.Thetheory ofcoincidences

3.Photonsandbosonsinachaotic state:permanental pointprocesses

4.Electronsandfermionsinachaotic state:determinantal pointprocesses

5.Thesymmetry between chaotic bosonsandfermions

1.Thecontext ofresearch inphysics around my thesis

1954sq Studies ofAndréBlanc-Lapierreaboutrandom fonctions

entrephotons:controverseentrephysiciens1960Découvertedulaser

NoticedeBernardPicinbono

AndréBlanc-Lapierrearenown physicist andacademician

- he left his markbyintroducing inFrancethethen emerging informationtheory,- he extended it tooptics,- well ahead ofhis time,he seeked toimmerse Statistical Mechanics inarigorous probabilistic framework:◦ Early in1954he firstexplained properly thephenomenom ofopticalcoherence (interference fringes)with thetheory ofrandom functions andthenotionofcorrelation.His papers,solely inFrench,were ignored(knowingly?)

◦ 10years later, with thediscovery anddevelopment oflasers,this viewpointbecame universal.Andit was extended byRoyGlauberinQuantumMechanics.

1954ABLexplains optical coherence,aliascorrelation between lightbeam fluctuations

Applying tooptics his newtheory ofrandom functionsMathematical foundations:◦ ‘Theoryofrandom functions’A.Blanc-Lapierre,R.Fortet,Ed.Masson,Vol.11965,Vol.21968

Optics:◦ Alightsourcecreates anelectromagnetic field 𝑥 𝑀, 𝑡 inspace𝑀 attime𝑡:it is arandom function◦ Forincoherent lightthefield is aGaussian random function

Coherence – Correlation:◦ Two stationary lightsources:inanobservationpoint𝑀twosuperimposedfields𝑥1 𝑡 and𝑥2 𝑡 .Tocreate interferences (fringes),putadelay 𝜏 onsecondsourcepath

◦ Average lightintensity

TwissAtthesame time,thefirstopticalcoherence(interference)experience atthequantumlevel:thephotodetectiontimesforthetwolightbeamsappear correlated:abunching effect

1956

“Inthisopticalsystem,whatisfundamentalisthatthetimeofarrivalofphotonsatthetwophotocathodesshouldbecorrelatedwhenthelightbeamsincidentuponthetwomirrorsarecoherent.However,sofarasweknow,thisfundamentaleffecthasneverdirectlybeenobservedwithlight,andindeeditsveryexistencehasbeenquestioned”

HB&T

Theywereright,thiseffecthasneverbeenobservedwithcoherentlight:itcouldnot,sinceitoccursonlywithincoherent(chaotic)light.Butweknowinhindsightthatthemercurylightsourcetheyviewedascoherentwasinfactincoherent,somethingtheydidnotknowandcouldnotknowbecauselaserwasnotyetinvented!

Badearly reception ofthe1956HBTexperimentThisexperiment favors thewave theory oflightandseems tocontradict quantummechanicsandthephotontheory oflight.Itwas very puzzling forphysicists.Inthefiftiesit provoked aheated debate abouttheconceptofthephoton.

What is easily conceivable intheclassical wave formalism (e.g.with aradiowave)becomeshardtoimagineatthequantumlevel:why should photonscling toeach other?

Later(1991)HanburyBrownwrote« Tometheinteresting thing aboutallthis fuss was thatso many physicists had failed tograsp howprofoundly mysterious ligth really is,andwererelectant toaccept thepractical consequences ofthefact that modernphysics does notclaimtotelluswhat things arelike ‘inthemselves’butonly howthey ‘behave’…Ifoursystemwasreallygoingtoworkonewouldhavetoimaginephotonshangingabout,waitingforeachotherinspace!»

Infact HB&T became thefathers ofanewdiscipline« statistical optics »which investigatesstatistical laws forphotoncounting under various physical situations:

« Inmy opinionHB&T is moreaprecursor ofthequantumoptics effects involving photoncorrelation »AlainAspect,2009

Some Frenchresearch inphysics atthat timeinOrsay-1958:YvesRocard fromEcole Normale Supérieure whohelpeddeveloptheFrenchatomicbombbuildsupinOrsay thelinearacceleratorofparticlesforfundamuntal researchinphysics

-1961:AndréBlanc-Lapierre becomesthedirectoroftheacceleratorandbuildsupacollisionring(Anneau deCollision–ACO)

-1963:CollisionsbetweenelectronsandpositronsareobservedatACOforthefirsttimeintheworld.

-1965:BernardPicinbono,aformerdoctoralstudentofA.Blanc-Lapierre (inAlger)arrivesinOrsay,equippedwithexpertiseinthenecessarystatisticaltools:inoptics,itisnotpossibletotrackthefluctuationsoftheEMfieldcorrespondingtoahugenumberofasynchronouslightemittingatoms.

-1966:StatisticalopticslaunchedinOrsay.Bernardhadinvestigatedthedetectionofweakopticalsignals,embeddedinthebackgroung randomnoisecalled‘shotnoise’(bruitdegrenaille).AfterA.Blanc-Lapierre,hedeepensthestatisticsoffluctuatingphysicalphenomena.Helaunchesanewlab:‘Laboratoire d’études desphénomènes aléatoires’(LEPA),withresearchprograminstatisticalopticsandsignalprocessing.

-LEPAwaswelllocatedintheInstitut d’electronique fondamentale (IEF)whereexperiencesofopticalilluminationofparticleandlaserswereexperimentedwithpowerfulelectronicmicroscopedesigned!

BernardPicinbono andstatistical optics- BernardPicinbono wishes tounderstand why thebunching effect ofphotonsdisappears withmonomode(coherent laser)light.

- Heis convinced that theclassical (ABL)approach worksforlaserasit does forincoherent (natural)light,andthat it should explain why thebunching effect does notarisewith coherent light

- In1964attheprestigious FrenchÉcoledesHouchesdePhysiqueThéoriqueBernardmeets RoyGlauberwhowas presenting his quantumtheory ofopticalcoherence (2005NobelPrize)

After thelaserdiscovery RoyGlauberintroduces anewquantumformalismvalid forcoherent andincoherent beams:it explains optical coherence

Bernard’s Orsayteamofyoung scientists1967he hires ateamofresearchers tolaunch Statistical Optics inOrsay

experimenters inoptics (lasers):CherifBendjaballah,MartineRousseau-LeBerre

theoricians inmathematics andphysics:◦ Amathematician forthebunching effect ofbosons:

OdileMacchi,experienced inprobabilities will deepen thetherory ofpermanental processes◦ Aphysicist forfermions:

ChristineBénard,who chosethequantumformalism ofGlauber

Both collaborate forfermions:discovery ofdeterminantal processes

Clever experimental physicists

Asound experimental study (photocounting andintervalmeasurements) forthePPofelectrons emitted byacathodeimpinged onbyvarious typesoflightbeams,ranging from achaotic beam (thermallight,curve A),unto apurelaserbeam (coherent light,curve E),inconnectionwith thefluctuationproperties ofthelightintensity.Thedistributionofthenumber ofelectrons showsclearly thatthebunching effect is reserved tochaotic beams

2014:aGermanmathematician writes astrikingly humbleletter inFrenchtoOdileMacchi

‘Madame,m’autorisez-vousàtraduirevotrethèsefrançaise(200pages)pourpublicationsousformedelivreanglaisparmonami,l’éditeurallemandDr.WalterWarmuth?’ProfessorHansZessin,Berlin

in2017:thebookinEnglishHans:Whyhundreds ofnonrewarding hours forsuch awork?Walter:Why taking therisk ofediting abookwith atiny market andnoprofit?

Merely forthesake ofmaking truth known better!

‘Togive this seminal work theplaceit deserves,asacornerstone whichconnects QuantumOptics andmodernpointprocess theory’(Bookintroductionp.7)

Conclusion…ofmythesis(January12,1972)

My viewpoint:find theoretical tools matched tophysics andexperimentationGoal:usethese tools todeepen thestatistical study ofbosonsandfermionsFurther studies andgeneralizations:this is upto…YOU!- apply such tools toother problems andinparticular inthetheoreticalstatistics:- find newPPmodels thanks tothetheory ofcoincidences- enlarge thetheory to◦ abstractspaces◦marked orother moregeneral PP

2.Thetheory ofcoincidences:Indistinguishable occurrences

Validity in𝓧⊂ℝ𝒏Noassumption ofstationarity

Thecoincidence densitiesℎ@(t1 ,…,tk ) 𝑑𝑡F ...𝑑𝑡@ =Ε 𝑑𝑁 𝑡F …𝑑𝑁 𝑡@ =Ρr 𝑑𝑁 𝑡F = … = 𝑑𝑁 𝑡@ = 1

Theℎ@ aretheprobability densities that thePPhaskoccurrenceslocated intheinfinitesimal intervals 𝑡P, 𝑑𝑡P ,j =1,…,k, nomatter whether there areotheroccurrenceselsewhere ornotTheCDarethelimiting aspectofthejointcounting distributions.They arephysicallymeasurable localquantities,independent ofthevolume𝓧 where thePPis observed.Thus they allow extending thedefinition ofthePPover𝑎𝑙𝑙ℝY.They arenonnegative,bounded,symmetrical functions.They arealso thedensities ofthefactorial momentsofthe numbers ofpointsindisjointintervals.Notany sequence ofnonnegative symmetrical bounded functions (even properlynormalized)represents theCDofaPP.Thereareconditions.Foranonstationary Poissonprocess with intensity 𝜌 𝑡 ,the𝑑𝑁 𝑡^ areindependent:

◦ ℎ@(t1 ,…,tk )=𝜌 𝑡F … 𝜌 𝑡@ .

Theexclusiondensities𝑝Y(t1 ,…,𝑡Y)=𝑛!Pr 𝑁 𝑋 = 𝑛, 𝑑𝑁 𝑡F = … = 𝑑𝑁 𝑡Y = 1

The𝑝Y aretheprobability densities that thePPhasexactly noccurrenceslocated intheinfinitesimal intervals 𝑡P, 𝑑𝑡P ,j =1,…,n, exclusiveofanyother occurrenceelsewhereAny sequence ofnonnegative bounded symmetrical functions𝑝Y(t1 ,…,𝑡Y) normalized according to

defines auniqueregular PPhaving this sequence asED.Noother condition.However theEDarenotphysically measurableForaPoissonprocess with intensity 𝜌 𝑡

◦ 𝑝Y(t1 ,…,𝑡Y)=𝜌 𝑡F … 𝜌(𝑡Y)𝑒h ∫ j k lkm

ImportantquantitiesThegenerating function ofthetotalnumber Nofoccurrencesin𝓧g(𝑣)=E 1 − 𝑣 𝑁

Theprobability ofnooccurrenceintheobserved volume𝒳 isP 𝑁 = 0 =g(1)g(𝑣)expands ∑ −𝑣 𝑝t

uvw 𝐸(𝑁 u )/p!)which generates thefactorial moments,

𝐸(𝑁 u )=E (𝑁(𝑁 − 1)… (𝑁 − 𝑝 + 1))Thefact that g(𝑣)is thegenerating function ofanonnegative integer plays abasicrole inthetheoryThejointcounting factorial momentsinq disjointsubsets I1 ,I2 ,…,Iq of𝒳

M 𝑝𝑞 =E{𝑁FuF …𝑁~

u~ } areobtainedbymere integration ofℎuF�…�u~overI1,I2,…Iq

RegularPPLet𝒂𝒑betheconvergenceradiusoftheentireseries associated tothesystemofED

(((=∑ 𝑧Y�Y P 𝑁 𝑋 = 𝑛 )

Result 1if𝒂𝒑 > 𝟏theprobabilitylawofthePPiscompletelycharacterizedequivalentlybythesystemofCDorbythesystemofED.Thenthe systemofCDfollows fromthesystemofEDthrough thedirectrelationships

andthePPis called regular

Moreover 𝒂𝒉v 𝒂𝒑 - 1,where 𝒂𝒉is theconvergenceradiusoftheentire seriesΦ�(𝑧) similarlyassociatedtothesystemℎ@(t1 ,…,tk )ofCD

Completely regular PPAregular PPis said completely regular ifits systemofEDcan be derivedfrom its systemofCDthrough theinverserelationships

Result 2:if𝒂𝒑 > 𝟐 orequivalently 𝒂𝒉 > 𝟏 thePPis completely regular.

Result 3(basic):Coincidence based constructionofacompletely regular PPAsystemofnonnegative,bounded,symmetrical functions ℎ@(𝑡F, … , 𝑡@) isthesystemofCDofacompletely regular PPif𝒂𝒉 > 𝟏andifallthe𝐬𝐞𝐫𝐢𝐞𝐬𝒑𝒏(t1,…,𝒕𝒏)arenonnegative (thenormalizing conditionisautomatically satisfied).Unicity ofthis PPResult 4:If𝒂𝒉 <1thePPis notcompletely regular,onecannot derive theEDfrom theCD.Thereis nogeneral result if𝒂𝒉=1.

3.Photonsandbosonsinachaotic state:permanental pointprocesses

Cox(orConditioned Poisson)Processes:general regular caseDefinition:Aregular Coxprocess is aPPwhose CDarethemomentsofsome nonnegative function I(t)called theunderlying intensity:

ℎ@(t1 ,…,tk )=E 𝐼 𝑡F … 𝐼(𝑡@)

LetI be the bounded rectangleinℝ� where thePPis observed,andI(t)a nonnegative randomfunction.Then,given asample ofI(t),thePPis Poissonwith nonstationary intensity I(t),because theform 𝐼 𝑡F … 𝐼(𝑡@)ofits conditional CDis characterisitic ofaPoissonprocess.Thus physicists callthisPPaconditioned Poissonprocess.

Result 5Existencetheorem:Ifanonnegative random function I(t)is such that

E exp𝑎 ∫ 𝐼 𝜃 𝑑𝜃�

< ∞, forsome𝑎 > 0,

thereexistsauniqueregularCoxProcesswithunderlyingintensity𝐼(𝑡) onI.Its CDareasabove anditsEDare 𝑝Y(t1 ,…,𝑡Y)= E 𝐼 𝑡F … 𝐼(𝑡Y)exp𝑎 ∫ 𝐼 𝜃 𝑑𝜃�

Ifmoreover𝑎 > 1, thisCoxprocessiscompletelyregular.

Result 6:We haveproved theexistenceofCoxprocesses whose underlying intensities I(t) assumesometimes negative values.Thisis possiblee.g.with aGaussian I(t) whose (positive)mean valuem(t)dominates the(positive)covariancefunction inthesense that m(𝑡) ≥ ∫ 𝐶 𝑡, 𝑢 𝑑𝑢£ a.e.

1965:Detecting photonsofalightbeamageneral CoxProcess

Rev.Mod.Phys. 37,pp.231-287,1965 “Coherencepropertiesofopticalfields ReviewsofModernPhysics”MandelL.andWolfE.:with both theclassical andquantummechanicsformalisms,they showed that attheoutputofanoiseless photodetecting surfaceimpingedonat(𝑡 = time𝜃, place𝑥 )byapartially polarized weak beam oflight(natural orlaser),thePPis aCoxprocess whose underlying intensity is the(random)lightintensity :

ℎ@(t1 ,…,tk )=E 𝐼 𝑡F … 𝐼(𝑡@) ; 𝑝Y(t1 ,…,𝑡Y)= E 𝐼 𝑡F … 𝐼(𝑡Y)exps ∫ 𝐼 𝜃 𝑑𝜃�

where s is thedetectorefficiency,I is thebounded rectangleinℝª where thePPisobserved, X(t) thecomplex analytical signalassociated totherealelectromagnetic fieldE(t);andI(t) = s 𝑿(𝒕) 2

Therearedifferent Coxprocesses forvarious statistical properties ofE(t)(thus ofI(t)).Cf inparticular themany papers ofB.Picinbono’s teamintheyears 1968- 1975.Perfect laser:only thephaseofE(t)fluctuates,I(t)nonrandom, thePPis nonstationaryPoisson.

1970- ChristineBénard’smodel:Beams ofquantumparticles with thewave packet formalism

Extending thework done atsecondorder byM.L.Goldberger,1963,sheconsideredthecoincidencesofallorders:𝐢nafinite rectanglecavity I,arandom numberN ofnoninteracting andindistinguishable particles 𝑸𝒊 ,either bosonsorfermionsaresuperimposed. 𝑄^ isdescribedbyitsrandom‘wavepacket’∅ 𝒕𝒊 , 𝑡𝑖 =time𝜃𝑖, location𝑥𝑖 .Thisisanunobservablemodel,nodetetionconsidered.

Thewavepacketsareindependentofoneanotherandhavecovariancefunctiondenoted f 𝒕, 𝒖 = 𝐸 ∅ 𝑡 ∅ ∗ 𝑢 .Meansarequantummechanical averages.

Thebeam (EMfield)is described byits random ‘wave function’,superposing allwave packets byappropriate projections

☼ onasymmetrized space forbosons(integer spin)

☼ onananti-symmetrized space forfermions(spin=odd half-integer).

TheEMfield covariance,C(t,u), follows from thewave packets covariance

Buttheresut is intractable unless thebeam hasweak density

Even then it is neat forbosons,butnotforfermions.

Phys.Rev A,vol2,n° 5,2140-2153,Nov.1970« FluctuationsofBeams ofquantumparticles »

Thepermanental modelforachaotic beam ofbosonse.g.photonsofathermallight

Result 7:Forchaotic bosonstheCDarewrittensolelywiththecovariance ofthewave functionC(𝑡^,𝑡P) = 𝐸 𝑋 𝑡^ 𝑋∗(𝑡P) (notnecessarily stationary).Theyarethepermanents

hk(t1 ,…,tk ) =∑𝑃¼ ∏ 𝐶(𝑡^, 𝑡¼^@^vF

�� )ofthematrices

𝐶(𝑡F, 𝑡F) ⋯ 𝐶(𝑡F, 𝑡@)⋮ ⋱ ⋮

𝐶(𝑡@, 𝑡F) ⋯ 𝐶(𝑡@, 𝑡@)

∑ 𝑃𝛼�� meanssummingoverallpermutations(𝛼F, …𝛼𝑘)of(1, … ,k)

Why?Because thefield E(t)is azero-mean Gaussian random process.Its analytical signalX(t)(nonegative frequencies)is acomplex, zeromean, stronglyGaussianprocesswithE 𝑋 𝑡^ X(𝑡P) =0.

Result 8(existence):Thereexists a(unique)PPwith theabove CDTheproofuses 𝜑𝑖(𝑡) ,acomplete systemoforthonormal functions onI,that areeigenfunctions ofC(t,u)

𝜆𝑖𝜑𝑖(𝑡)=∫ 𝐶(𝑡, 𝑢)� 𝜑𝑖(𝑢)du𝜆𝑖>0, ∑ 𝜆𝑖�^ < ∞, ∫ 𝜑𝑖(𝑡)𝜑P∗(𝑡)� dt =𝛿^,P

andtheKahrunen-Loeve expansionC(t,u)=∑ 𝜆𝑖𝜑𝑖(𝑡)𝜑^∗ 𝑢�^ ,

toshowpositivity oftheCD,andvalidity ofconditionE exp𝑎 ∫ 𝐼 𝜃 𝑑𝜃�

< ∞,e.g.for𝑎 < 1/(2𝜆ÈÉÊ)

Theexclusiondensities themselves arepermanents

Result9 Thegenerating function forthedetection ofa chaotic bosonbeam isE 1 − 𝑣 𝑁 =g(𝑣)=∏ 1/(1 + 𝑣𝑠𝜆𝑖)t

^vFWith thehelpofthefunction 𝑓 𝑡, 𝑢 relatedtothefield covarianceC(t,u)through

andwith thepermanentsofthematrices𝑓(𝑡F, 𝑡F) ⋯ 𝑓(𝑡F, 𝑡@)

⋮ ⋱ ⋮𝑓(𝑡@, 𝑡F) ⋯ f(𝑡@, 𝑡@)

Then theEDare pn(t1 ,…,tn ) =sn h(s)∑𝑃¼ ∏ 𝑓(𝑡^, 𝑡¼^Y^vF

�� ) h(s)=g(1)

Moreover f(t,u)=∑ (𝜆𝑖/(1 + 𝑠𝜆𝑖))𝜑𝑖(𝑡)𝜑^∗ 𝑢�^

which evidences that f(t,u)is acovariancefunction andthus that these EDareindeed positive

Physically itturnsoutthatf(t,u) = 𝑬 ∅ 𝒕 ∅ ∗ 𝒖 𝐢𝐬 thewave packets’ covariance

Consistency

C(t,u) is thecovariancefunction ofthewave function,X(t),thecomplex randomelectromagnetic field

Thus C(t,u) is positivedefinite

Thisfact is ofcritical importancefortheconsistency ofthePPmodel.

Through thepositivity oftheeigenvalues 𝜆^,this fact implies that f(t,u)is also positivedefinite.

Inturn positivedefiniteness off(t,u)controls thepositivity ofthefunction h(s)andofthepermanental expressionsoftheCDandED.

Positivedefiniteness off(t,u)also controls the(existence)sufficientconditionE expÉ ∫ £ k lkÎ < ∞forsome 𝑎 > 0.

Physicalconsiderations showthat f(t,u)is thecovariancefunction ofthewave packets

Thebunching effect ofthepermanental PP.

Result 10:Bunching effect :

𝒉𝟐(t1,t2 )=𝒉𝟏(t1) 𝒉𝟏(t2 ) +s2 |𝑪(t1,t2)|2 >𝒉𝟏(t1) 𝒉𝟏(t2 )

OrequivalentlyPr{dN(t2)=1/dN(t1)=1}=Pr{dN(t2)=1}+s|𝑪(t1,t2)|2 /C(t1, t1)

Theaposterioriprobability todetect any photonatagiven time,when anotherphotonhasbeendetected inaneighboring time(secondorder coincidence)ishigher than theaprioriprobability todetect aphotonatthat time:two photonstendtoagregate!

Chaotic photonsandbosonsbehave like sheeps!

Stationary (time)case:C(t +d,t)=𝛤 𝑑 E(N)=T I̅

Lorentzspectrum: 𝜏 = coherencetimeofthelightfield

𝛤 𝑑 = I̅exp(− 𝑑 /𝜏 )

Bunching effect:𝐵2(d)=𝒉𝟐(t1 ,t2 )/ 𝒉𝟏(t1) 𝒉𝟏(t2 )

=1+exp (– 2 d /𝝉) (1:noagregation)

𝛾 𝜈 = 2I̅ 𝜏 /(1+4𝜋2𝜏2𝜈2)

𝜈𝜏

d/𝜏

Higher order bunching effects ofphotons(Lorentzian spectrum light)

Threepointst1 ;t2 =t1+ d1 ;t3 =t2+d2 ,d1 >0;d2 >0If𝐵3(d1,d2)=ℎÛ(t1 ,t2 ,t3)/ ℎF(t1) ℎF(t2 )ℎF(t3)B3(d1,d2)=𝐵2(d1) +𝐵2(d2)+B2(d1+d2)+ 2 exp – 2(d1+d2)/𝝉)Thebunching factorbetween three occurrencesis superior tothesum ofthethree pairwise bunching factorsTheagregation tendency is reinforced asthenumber ofoccurrencesincreases.

4.Electronsandfermionsinachaotic state:determinantal pointprocesses

Avery longgermination…

Feb.22,2013EtienneGhys toOdile:Onthis board,is it you« ThéorèmedeMacchi1975 »?OdiletoEtienne« Certainly not! »

Here AlexanderBufetov,oftheLyon1University,atanLATP2013Colloquium liketheoneoftoday

1973:Thedeterminantal modelforabeam offermionsAgainChristineBénard,butwith OdileMacchi:J.Math.Ph.,vol14,n° 2,155-167,Feb.1973 « Detectionand‘emission’processes ofquantumparticles inachaotic state».With thewave packet formalism weconsidered arandom number N ofnoninteracting andindistinguishable fermions(spin=odd half-integers)inarectanglecavity:electrons,protons,neutrons…

Theparticle 𝑄^ found in𝑡^isdescribedbyits random wave packet ∅ 𝑡^ , notnecessarily real,butindependent oftheother wave packets.Therandom wave function ofthefermionbeam follows byappropriate projectiononananti-symmetrized space.Means arequantummechanical averages withrespecttothefield operator. Then thevery intricate expressionoftheCDinvolvesthecovarianceC(t,u)ofthewave function.

Forachaotic beam,theCDreduce tothedeterminants ofthecovariancematrix

hk(t1 ,…,tk ) = det𝐶 𝑡𝑘 = ∑𝑃¼(−1)∏ 𝐶(𝑡^, 𝑡¼^@^vF

�� )

where C{tk} =𝐶(𝑡F, 𝑡F) ⋯ 𝐶(𝑡F, 𝑡@)

⋮ ⋱ ⋮𝐶(𝑡@, 𝑡F) ⋯ 𝐶(𝑡@, 𝑡@)

∑ 𝑃𝛼(−1)�� meanssummingoverallpermutations(𝛼F, …𝛼𝑘)of(1, … ,k),each term affected with the

sign(−1)Ý ¼ , 𝑟(𝛼)denoting thesign ofthepermutation(𝛼F, …𝛼𝑘).

Existenceofdeterminantal PPResult 11,existenceofDPP:The necessary andsufficient conditionsforaseries ofbounded,symmetrical,nonnegative functions ℎ@(t1 ,…,tk ) ofthedeterminantal form tobe theCDofaregular PPonIare

Condition1.Thefunction C(t,u)onwhich thefunctions ℎ@(t1 ,…,tk ) arebased is positivedefinite.

Thisconditionis necessary andsufficient fornonnegativity ofalltheℎ@(t1 ,…,tk )Condition2. 𝝀𝒊 < 1foralli.

Undercondition1there exists acomplete systemoforthonormal functions 𝜑𝑖(𝑡) onIthat areeigenfunctions ofC(t,u)

𝜆𝑖𝜑𝑖(𝑡)=∫ 𝐶(𝑡, 𝑢)� 𝜑𝑖(𝑢)du∫ 𝜑𝑖(𝑡)𝜑P∗(𝑡)� dt =𝛿^,P,

andsuch that C(t,u)=∑ 𝜆𝑖𝜑𝑖(𝑡)𝜑^∗ 𝑢�^ ,

with thebasicproperties that 𝝀𝒊>0, ∑ 𝝀𝒊�𝒊 < ∞

Proofofcondition2Anecessary condition:Ifthemodelis consistent

g(𝑣)=∏ (1 − 𝑣𝜆𝑖)t^vF

is thegenerating function ofthenonnegative,integer number N offermionsThisrequires that 𝝀𝒊 ≤ 𝟏, ∀𝒊

andeventhat𝝀𝒊 <1, ∀𝒊

( 𝜆^w =1forsome i0would yield alltheED=0)Asufficient condition: Assumethat 𝝀𝒊 < 𝟏, ∀𝒊

Then theinversionformalism is valid andyields thefunctions

pn(t1 ,…,tn ) =∏ (1 − 𝜆𝑖)t^vF ∑𝑃¼(−1)∏ 𝑓(𝑡^, 𝑡¼^

Y^vF

�� )

where f(t,u)is theresolvent oftheFredholmequation:

f(t,u)- ∫ 𝑓 𝑡, 𝜃 𝐶 𝜃, 𝑢 𝑑𝜃 = 𝐶 𝑡, 𝑢 𝑡, 𝑢 ∈ I�

Itis worth f(t,u)=∑ (𝜆𝑖/(1 − 𝜆𝑖))𝜑𝑖(𝑡)𝜑^∗ 𝑢�^ ,andthus is positivedefinite.

Therefore thepn(t1 ,…,tn )are nonnegative,they areindeed theED

Physicalinterpretation:thePauliexclusionprincipleInthewave packet formalism,denote 𝜓@(t)acomplete systemoforthonormal modesforthe(bounded)cavity I,nk thenumber offermionsinmodek, 𝑛𝑘 its average.

Physicalconsiderations showthat thewave packet covarianceg(t,u)= ∅ 𝑡 ∅ 𝑢 reads:

g(t,u) =∑ ( 𝑛@@ /(1- 𝑛@ ))𝜓@(t)𝜓@∗ 𝑢

Identificationwith our modelLet𝜆@= 𝑛𝑘 betheaverage numbers offermionspermode

𝜑𝑘(𝑡) = 𝜓@ (t)bethemodesofthecavityI

thenour𝑓 𝑡, 𝑢 = ∑ (𝜆𝑖/(1 − 𝜆𝑖))𝜑𝑖(𝑡)𝜑^∗ 𝑢�^ isthewave packet covariance

Clearly the𝝀𝒊 mustbe positive(condition1):they arethemean number offermionspermode

Andthe𝝀𝒊shouldbeless than 1(condition2): this is aproperty specific offermionbeams:

‘Atmost onefermionperquantummode’,i.e.thePauliexclusionprinciple

C(t,u)=∑ 𝜆𝑘𝜑𝑘(𝑡)𝜑@∗ 𝑢�@ becomesthe(quantum)covarianceofthewave function (field).

Theanti-bunching effect ofthedeterminantal PPResult 12:Anti-bunching effect :

𝒉𝟐(t1,t2 )=𝒉𝟏(t1) 𝒉𝟏(t2 )- |𝑪(t1,t2)|2

𝐵2(t1,t2)=𝒉𝟐(t1,t2 )/ 𝒉𝟏(t1) 𝒉𝟏(t2 )=1- |𝑪(t1,t2)|2 /(𝒉𝟏(t1) 𝒉𝟏(t2 ))<1

(1:noexclusion)

𝐵2(t,t)=0absolute exclusion

Theaposterioriprobability todetect any fermionatagiven time,whenanother fermionhasbeendetected inaneighboring time(secondordercoincidence)is smaller than theaprioriprobability todetect afermionatthattime:two fermionstendtoexclude each other!Athigher orders thedeterminantal expressionsof CDexhibit similar exclusionproperties.

Chaotic fermions(electrons)behave like foes!

Example:ThetimeDPPwith Lorentzian properties1.Thestationary case:LorentzspectrumC(t +d,t)=𝛤 𝑑 = I̅exp(− 𝑑 /𝜏 ):𝐵2(d)=1- exp (– 2 d /𝝉)

2.Thenonstationary case:generalized renewal PPC(𝑡F, 𝑡Û) C(𝑡Û, 𝑡ä) = C(𝑡F, 𝑡ä) C(𝑡Û, 𝑡Û) with t 1 ≤t2 ≤t3LetD(t,u) be thenormalized wave covariance:C(𝑡, 𝑢) =D(t,u) C 𝑡, 𝑡 C(𝑢, 𝑢)�

hn(t1 ,…,tn) = ∏ C(𝑡^, 𝑡^)Y^vF ∑𝑃¼(−1)∏ 𝐷(𝑡^, 𝑡¼^

Y^vF

�� )

hn(t1 ,…,tn) =∏ YhF^vF (1- 𝐷 𝑡^, 𝑡^�F 2)∏ ℎ1(𝑡^Y

^vF )with successivetimest 1 ≤…≤t3Thischaracterizes generalized renewal:

intervals between successiveoccurrencesareindependent butnotequidistributed

0,0000

0,2000

0,4000

0,6000

0,8000

1,0000

1,2000

-4 -3 -2 -1 0 1 2 3 4

B2 (d)

|𝑑|/𝜏

5.Thesymmetry between chaotic bosonsandfermions

Which experimental results?Bosons:Attheepoch we wrote our paper (1969-1972)thelaserwas invented andmuch could be done experimentally toevidence thebunching effet.Other bunchingeffects havebeenobserved with lasers,according towhich partofthefield E(t)fluctuates (e.g.only thephase,then I(t)nonrandom,thePPis nonstationaryPoisson).

Howis thebunching effect observed now?Hasabunching effect higher than forchaotic photonsbeenobserved ?What aboutbosonparticles other than photons?

Fermions:Atthis epoch allexperimental fermionsources,even thebestmonocinetic andpowerful ones (point-cathodeelectron sources)could notprovidecoherence timeslarger than 10-13 sec.,while electronic detection devices involvedintegration overtimesontheorder of10-9 sec.much larger than thecoherencetime.Therefore our paper was purely theoretical.

Howtheanti-bunching effect hasit beenobserved now?forelectrons?