Exploring Euclidean Path-Integral Approach in Black Hole Information Theory
Delve into the concept of black hole information emissions through quantum extremal surfaces, the subtleties of interpreting path-integrals, and embedding images into orthodox path integral formulations. Follow the Euclidean path-integral approach, analyze steepest-descent approximations, and explore the possibility of explaining the Page curve using this method.
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Presentation Transcript
Yang-Millsinstantonsas theendofblackholeevaporation SasakiandDY,1404.1565 Chen,SasakiandDY,1806.03766 Chen,SasakiandDY,2005.07011 Chen,Sasaki,DYandYoon,2111.01005 Chew,Chen,SasakiandDY,inpreparation Dong-han Yeom Department of Physics Education, Pusan National University 2023. 8. 1.
Howcanablackholeemitinformation viaquantumextremalsurfaces?
Theinterpretationissubtlebecause thisisapath-integralaboutnot densitymatrices states but
Howcanweembedthispictureintothe orthodox path-integralformulation?
LetusfollowtheEuclideanpath-integralapproach. (Hawking,2005)
??|?? |? = ? ? ? = ? ????????? path integral as a propagator
??|?? |? = ? ? ? = ? ??????? ?? Euclidean analytic continuation
??|?? |? = ? ? ? = ? ??????? ?? ?? ????? ? ??? ?? steepest-descent approximation need to find/sum instantons
|?? ? ? = ? ??????? ?? |?
|?? |?? ? ? = ? ??????? ?? ??|?? |? = ? |? Hartle and Hertog, 2015
|?? |?? ?? ????? ? ? ? ??? ?? ??|?? |? = ? Lorentzian Euclidean Lorentzian |?
CanweexplainthePagecurve usingtheEuclideanpath-integralapproach?
Indeed,thecomputationsofthePagecurvearejustified bythefollowingtwosteps.
Indeed,thecomputationsofthePagecurvearejustified bythefollowingtwosteps. First,thereareatleasttwohistories theentanglemententropy, whereone (say, 1) theother (say, 2) thatcontributeto isinformation-losingwhile isinformation-preserving.
Indeed,thecomputationsofthePagecurvearejustified bythefollowingtwosteps. First,thereareatleasttwohistories theentanglemententropy, whereone (say, 1) theother (say, 2) thatcontributeto isinformation-losing isinformation-preserving. while ? ?1?1+ ?2?2
Indeed,thecomputationsofthePagecurvearejustified bythefollowingtwosteps. Second,theprobabilityoftheinformation-preservinghistory dominantatthelatetime. Then,onecanreproducethePagecurvethatwewanted. ? ?1?1+ ?2?2
1isinformation-losing. AandBdenotetunneling: theycanhappenatanytime.
2sareallinformation-preserving. 1isinformation-losing. AandBdenotetunneling: theycanhappenatanytime.
2sareallinformation-preserving. 1isinformation-losing. (2) isdominatedatlatetime. 2 AandBdenotetunneling: theycanhappenatanytime.
Tosummarize, ifweassume(1)multi-historycondition and(2)late-timedominancecondition, onecanexplainthePagecurve.
??????? ? ? ? ??? ?? (I) (III) cusp ? = ??/2 ? = 0 (II) Even Hawking radiation can be interpreted as instantons of a free scalar field. (Chen, Sasaki and DY, 2018)
Aftersomecomputations,finallywecanrecoverHawkingsresult. ? 2? ? 8????if ? ? furtherobservethatthereexistplentyofinstantons with??/? ? We Inthelimit?? = ?,weobtainthetrivialgeometry, whereonecanguaranteetheexistenceofinstantons.
Thereexistsatunnelingchanneltowardatrivialgeometry thankstotheplentyofinstantons.
Forthefirsthistory,weassumethattheentanglemententropy monotonicallyincreases.
However,sincethereisnointeriorforthesecondhistory, theentanglemententropy (betweenblackholeandradiation) iszero.
Astheblackholesizedecreases, thecontributionfromthemultipleperiodsbecome moreandmoreimportant.
Theprobabilitytotunneltoatrivialgeometryis dominatedatthe (very) latetime (aftersomecomputations).
? ?1?1+ ?2?2 Therefore,onecanmimic aPagecurve, whilethisisnothingtothewiththequantumextremalsurfaces, butonlyrelyingontheEuclideanpathintegral.
? ?1?1+ ?2?2 Oneinterestingpointisthatthisallowsamoment whentheBoltzmannentropyisgreaterthanitsarealentropy. Thismightbeasmallremnant oramonster.
CanthisbiasofthemodifiedPagecurve bechangedthankstomoreeffects?
WecanconsiderSU(2)Yang-Millsinstantons. Thenon-trivialsolutionisas theBartnik-McKinnonsolution.
Introducingmetric/fieldansatz, onecanderiveasetofequationsofmotion.
NumericalsolutionsareasymptoticallySchwarzschild, whilethecenterisregular.
Thereisaninfinitesequenceofsolutions, buttheirvolumeexponentiallyincreases. Therefore,itisreasonabletoconcludethat theonlyseveralnumberofsolutionswillcontributemeaningfully.
Therefore,Yang-Millsinstantonsprovide quitemany information-preservinghistories.
? ?1?1+ ?2?2 Ifweincludemoreinstantons, thiscurvemightbemodified. However,probably,itisinevitabletoviolatetheentropybound. Weneedmorecarefulcomputations.
Also,wereliedonEuclideaninstantons,but isthereanyexplicittime-dependentinstantons?
< 0 Bosonstarsmightbeanexample,but whyistheprobabilitynotbehavingwell? Somethingtodowiththeglobal chargeconservation?
Thankyouvery much
