4D SIMPLICIAL QUANTUM GRAVITY

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Arron Hubbard
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1 T.YUKAWA and S.HORATA Soknda/KEK D SIMPLICIAL QUATUM GRAITY Plan of th talk Rvw of th D slcal quantu gravty Rvw of nurcal thods urcal rsult and dscusson

2 Whr dos th slcal quantu gravty stand? In short dstanc xstnc of th contnuu lt In long dstanc Enstn gravty as th classcal lt

3 n-slx 3 D slcal anfold Condton : two -slcs attach through on 3-slxfac and on 3-slx s shard by two - slcs. Condton : -slcs sharng on - slx consttut -ball. x. sallst -ball

4 anfold condtons gv Eulr rlatonsh χ χ : total nubr of -slcs : Eulr charactrstc Choos and 3 χ as ndndnt aratrs

5 Slcal gravty Rgg96 Enstn-Hlbrt acton Eucldan S G { d x π g R n + n! + Λ d x cos g } + Λ{ n n a n volu of n-slx + π 5 cos π.99 }

6 Flds on slcal anfold Lattc -for flds L -for fld -for fld -for fld φ φ j φ jk on a st φ x on a lnk j x x j dx µ φµ x µ ν on a trangl jk dx dx φ x Dual flds on th dual lattc d-cll : d-olygon dual to a st Ld t jk µν d--cll j: fac shard by d-cll and d-cll j j

7 Dual flds ψ ψ j -for dual fld: on th d-cll -for dual fld: on th d--cll j -for dual fld: ψ jk on th d--cll jk Scalar roducts φψ Dualty rlatons φ l jk... L ψ jk... l j L φ jψ j d ψ jk... l... φ jk... jk σ d ψ a φ < φ ψ > σ jk... φ jk ψ jk t jk : volu of th slx jk : volu of th cll jk

8 Mass trs M φ < φ ψ Drvatvs d d : L L + rark: d Kntc trs K φ M dφ < dφ dψ > > M M σ φ l j σ j φ j φ φ j dφ j dφ jk φj + φ jk + φ k K σ j φ φ j l j K σ jk φj + φ jk + φk t jk

9 Mattr flds S M rarks: n X n A n K K n K X + K n A X ο lj X X j l j X ο tjk Aj + Ajk + Ak t jk d a σ a ο s d d Thn for ass trs σ a ο s a { s } n { s } and for kntc trs σ { s } a d

10 urcal thods Partton functon Z DX DA T S G M S S + S G X S A S Rark: s th su ovr all th ossbl gotrcal confguratons T Mont Carlo ntgraton Gnraton of a Markov chan α α... α α +... M S α TXA wth th wght Dtald balanc Sα Pα α n n α S α+ + + α + P α α α S S G + SM α n α : # of ossbl ovs fro α

11 Gotrcal ovs Elntary q-ovs q Rark: Ergodcty of lntary ovs

12 5 5 Mov Mov 33 Mov

13 Choc of th gotrcal ov Startng a nu confguraton wth a fxd toology Choos th -ov wth th robablty P S G js j G Hat-bath thod 3 q S G + +

14 Gotrcal constrants Aftr { }-ovs Bsds thr nvarant condtons for bcaus w hav startd a th nu confguraton 5 3

15 Postvty of th slcal acton Th two nqualty rlatons gv 5 5 Postvty of S G s satsfd Whn or + Λ cos 5 π π whch s always tru for. Λ -.98

16 Gotrcal ov contnud 3Scfy th ov by choosng a slx for th -ov randoly. Chck th anfold condtons. T T Acct th ov wth th robablty Mtrools thod } n{ T T n n T T P whr n T s # of ossbl ovs fro confguraton T: 5 T s T n T s Thn T T n n T T

17 Mattr ovs 5 Put flds randoly whn a nw vrtx and/or lnks ar addd. 6Mattr ovs ar ad for all X and A flds by th hat-bath thod. P S x X x x + S x S X dx X x ο lj x X j j 7Instad of x w ay choos Th ovr rlaxaton thod X x + X X j ο l j j ο l X j j

18 Rsults of th nurcal sulatons Calculaton of artton functons + A X T S M Z α S M α S M z α c ζ Wth a fxd toology S and n X n A.

19 c S W calculat z G S G + by tunng as. c wth a odfd acton

21 Scalar curvatur < >/ < d d > ln z c For larg t s boundd.. < >.5 nar.5: 5 ovs donat branchd olyr has nar.: ovs donat cruld surfac has

23 3 Curvatur susctablty χ ln χ d d z c > > < <

24 Short dstanc corrlaton n t C t { ο t ο n t t s a trangl havng a coon lnk wth th trangl t t }

25 Dscussons Is th analogy to th D gravty OK? In D th scalng rlaton of DDK holds xactly: z DDK αω zddk αω + Q In th DT calculaton w xct Z DT z DT T ζ T ωχ Thn w hav a corrsondnc: Z DT a ζ a a a λ dz a DT a c c c λ

26 In D th conforal gravty assus th scalng ωχ αω βω αω b C C z z Whl n DT w calculat S DT M Z α ζ α z S DT c M Th contnuous lt s takn as a DT c a a a a Z ζ DT z d λ ωχ αω βω αω ζ ζ b Thus

27 Th artton functon xctd fro th conforal gravty Z DT ϕ µ ν c whr α β ar solutons of α α β b β b wth b n X + 6nA Ths fors do not allow us to ak us of th old MIBU algorth and w hav to chck all th old rsults.

Grand Canonical Ensemble

Grand Canonical Ensemble Th nsmbl of systms mmrsd n a partcl-hat rsrvor at constant tmpratur T, prssur P, and chmcal potntal. Consdr an nsmbl of M dntcal systms (M =,, 3,...M).. Thy ar mutually sharng th total numbr of partcls