We start with a simple remark about amplitudes

Transkrypt

1 We start with a simple remark about amplitudes

2 The 1à 1 amplitude in String Theory

3 <latexit sha1_base64="cgn1xme4iujxliqnuil7xig0evc=">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</latexit> <latexit sha1_base64="x5bt+wee59aq2gm21znbzwpmvna=">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</latexit> The 1à 1 amplitude in String Theory A 1!1 = = = Z Z Z DX d 2 zv d 2 zv e S = Z vol(psl(2)) DX S V (0)V (1)e R vol(r U(1)) dx0 R (2 ) D 1 ( ~ k 1 ~ k2 )hv (0)V (1)i 0 d Energy is automatically conserved if we have on shell operators V. X 0 = 0 k 0 <latexit sha1_base64="jwbfl/j3vzcjor6673paqowx1r0=">aaacbxicbvdlssnafj3uv42vqmtublvrvunqqjdc0y3lcvybtsw300k7dpjgziku0iurp8wvocbu/qhx/o3tngttpxdhzdn3mvcep+fmktv+ngorq2vrg8vnc2t7z3fp2j9oytgvhdzjzgpr8ufszilavexx2kkehddnto2prqd++4ekyeloto0t6ouwifjaccgt9awsrudovy0vsqs8gcijhumxqydflz5vtqv2dhizodkpoxynnvxl9moshjrshiouxcdoljebuixwojhdvnieyaggtktpbcgvxjy7yokptdlhqsx0rqrp1n8tgyrsjknfd4aghnlrm4r/ed1ubrdexqikvtqi84+clgmv42kium8ejyqpnqeimn4vkyeiiernzpo6bwfx5mxsqlwds2rttlaux+v5ffejhaft5kbzvec3qigaikbh9ixe0zvxzlwy78bhvlvg5doh6a+mzx+cgju+</latexit> A 1!1 =2k 0 (2 ) D 1 D 1 ( ~ k 1 ~ k2 ) Similar story for two point functions in AdS.

4 Now to the topic of the talk

5 Symmetries Near the Horizon Juan Maldacena Institute for Advanced Study

6 Based on work with: Henry Lin Ying Zhao

7 Preliminaries

8 Black holes and quantum systems

9 Basic Assumption (central dogma) A black hole seen from the outside can be described as a quantum system with order S degrees of freedom (qubits) (coupled to the rest of spacetime) = S = Area 4G N T Hawking

10 Geometry of a Black Hole made from collapse Singularity Oppenheimer Snyder 1939 interior horizon star One exterior, one interior.

11 Full Schwarzschild solution ER singularity Eddington, Lemaitre, Einstein, Rosen, Finkelstein, Kruskal Left exterior Right exterior Vacuum solution. No exotic matter. Two exteriors, sharing the interior.

12 If one black hole = quantum system, What do these two connected black holes correspond to?

13 Wormhole and entangled states Connected through the interior = W. Israel J.M. Entangled In a particular entangled state

14 <latexit sha1_base64="hai71xbxn/jywxrl50j5pfn4yge=">aaacd3icbvdlsgmxfm3uvx1foy7dxgwldwgzgre6llrpsoj9qkeutjppqzmpkoxqhn6bkz/flaagbl278m9m21lo9udc4zxzse7xe86ksu0vo7c2vrg5vdw2d3b39g+sw6o2jfnbaivepbzdh0vkwurbiilou4mgopq57fitm7nfuadcsji6u9oe9km8iljacfzaglivcgaetxugl9y5cmflgmygco2bghn98zmod6ysxbmxgl/eyukj5wgore9vgjm0pjeihevzc+xe9tmsfcoczkwvlttbzijhtkdphemq+9linrlutdkeibb6raow6s+jdidstknfj0osxnlvm4v/eb1ubvf9jevjqmhelg8fkqcvw7wbgdjbiejtttartp8vybgltjru0dr1c87qzn9j2605fzx31i3vr/m+iugenaiqctalqqmgaqiwiugbpaex9go8gs/gm/g+jbamfoyy/ylx8q1z15jq</latexit> The near horizon region Left exterior Right exterior TFDi = X n e E n /2 Ēni L E n i R Boost = exact symmetry = 2 (H r H l ) (No simple bulk lattice discretizations in gravity)

15 Two other approximate symmetries X 0 X 1 Left exterior Right exterior E = global time translation symmetry P = Spatial translation X 0! X 0 + constant X 1! X 1 + constant <latexit sha1_base64="vrnju2j0wuridadq3knxdqe6mgk=">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</latexit>

16 This talk will be about these two other (approximate) symmetries X 0 X 1 They move us behind the horizon They move us from one side to the other Left exterior Right exterior

17 We will discuss this for near extremal black holes. These are described by Nearly-AdS 2 gravity. A similar structure appears in the SYK model. We will now review both cases.

18 Review of nearly AdS 2 gravity

19 Near extremal black holes M Q M Q N-AdS 2 x S 2 horizon

20 <latexit sha1_base64="tm2gil7n0tngd4nmoylp06b0vqu=">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</latexit> Nearly-AdS 2 gravity S = 0 applez Z R +2 K + Z (R + 2) + 2 b Z Jackiw-Teitelboim Almheiri-Polchinski K + S m [g µ, ] Only extermal entropy S 0 Fixes Metric to AdS 2 Matter moves in a rigid AdS 2 spacetime Boundary becomes dynamical = boundary graviton = only physical degree of freedom for gravity

21 The surprisingly simple gravitational dynamics of N-AdS 2 AdS 2 matter NAdS 2 = AdS 2 + location of boundary ds 2 = d 2 + d 2 sin 2 Dynamics of the boundary is SL(2) invariant. Proper time along the boundary = time of the asymptotically flat region = time of the quantum system

22 <latexit sha1_base64="9kacp4ku/gh4ueqlb86oj5/isrk=">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</latexit> AdS 2 ds 2 = d 2 + d 2 sin 2 matter Simplest case = Boundaries are infinitely far away. Matter moves effectively in all of AdS 2 and is not affected by the motion of the boundary. The boundary is still dynamical and tells us how to translate to the physical boundary time = time of the dual quantum system. Then the Hilbert space splits as: (H l H m H r )/SL(2) g Common motion of the three of them is not physical

23 <latexit sha1_base64="ivp3wu5qdp/mt4volge+1exf098=">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</latexit> Each of the three systems is described by an SL(2) g invariant action. The matter moves in a rigid AdS 2 Each boundary is like a massive particle with spin (or in an electric field in AdS 2 ) ab Y a Y b = 1 <latexit sha1_base64="izwppx92kh3q13yxvrxfxvek7yi=">aaacbnicbvc7sgnbfj2nr7i+vi21gewcnobdwggjbg0si5ihjgu4o5lnhsw+mjkvwplgyk+xehte1n+w8m+ctvjo4oelh3pu5d57vjgzqwz728gtla+sruxxzy3nre0da3evianeefoney9eywnjoqtpxthfasswfakp06y3vmr85gmvkkxhrrrf1a2ghzkfevba6lqhufjehaqgm4i3xnf3omvdfydoswh2ryjdtifai8szkqkaoda1vjq9icqbdrxhigxbswplpiaui5yozu4iaqxkch3a1jsegeo3nxwxxiwt9lafcv2hwhp190qkgzsjqf9zckan5lyxif957ut5527kwjhrncttrx7csypwfgnumugj4inngaimb8vkaaki0sgzwqro/m+lpfepo6flyk2lul2c5zfhb+gihsmhnaequky1vecepajn9irejcfjxxg3pqatowm2s4/+wpj8atvylys=</latexit> Embedding coordinates Rescaled embedding coordinates X a (u), X X =0 Ẋ Ẋ = 1 Proper time along the boundary = time of the dual quantum system.

24 <latexit sha1_base64="7yexsb9brj8eofzn2syuoy9bsgq=">aaaccnicbvdlssnafj34rpuvdelmsbeeosr1oruh6mzlc/ybbrom00k7dcyjmxohhg5d+smubavx6x+48m+cpflo64hhhs65lzv3+dgjutn2t7gyura+svnakm/v7o7tmwehbrklapmwjlgkuj6shngqtbrvjhrjqrd3gen4k9vm7zwqiwku3qtptfyorienkezks54jowvb2bwgj8fz2pt4aovv6hptw1bzmyt21c4bl4ltkaoo0pdmr/4wwgknocimsdlz7fi5krkkykzm5x4isyzwbi1it9mqcsldnl9kbk+1morbjpqlfczv3xmp4ljoua87ovjjuehl4n9el1hblzvsme4ucff8uzawqckyxqkhvbcs2fqthaxvf4v4jatcsodxzljwfm9eju1a1bmo1pq1sv2mykmejsejoamouar1cacaoauweatp4bw8gu/gi/fufmxbv4xi5gj8gfh5azzflfi=</latexit> SL(2) gauge constraint: Q a l + Q a m + Q a r =0 a = gauge index <latexit sha1_base64="2eylhwljpcdjwgmsfgroxr8ilvu=">aaacchicbvbns8naen3urxq/oh4fwwwftywpb70irs8ek9gpaevzbcbt0s0m7g7eeurjkz/fk6agxv0lnvw3jm0o2vpg4pheddpz3igzpw372ygsla+srhxxzy3nre0da3evqcjyumjqkiey7rifnaloaky5tcmjjha5tnzrvea37kaqfopbpy6gf5cbyd6jrkds3zre5tlgbf/gpcsdpcdxab4wex7ct8pls2+v7io9bv4ktk5kkee9b311vzdgaqhnovgq49ir7iveaky5tmxurcaideqg0empiagoxjl9y4kpu8xdfijtehpp1d8tcqmuggdu2hkqpvtzxib+53vi7z/3eiaiwiogs0v+zleocryk9pgeqvk4jyrklt6k6zbiqnuanzml4mz/veia1ypzwqneveu1yzypijpar+geoegm1da1qqmgougrpanx9gy8gs/gu/exay0y+cw++gpj8wdhapbt</latexit> Now one would be tempted to say that the symmetries we want are Q a m, the matter generators, since they are the ones that move the matter in AdS 2. But these do not have a translations to the boundary theory since they are not gauge invariant. So, we will write gauge invariant ones by gravitationally dressing them.

25 Short review of SYK

26 SYK Sachdev-Ye-Kitaev SYK: N Majorana fermions with all to all interactions. The theory has a simple large N limit, with an effective action which is a function of two times: G(u 1,u 2 ) <latexit sha1_base64="myu0hyye59cu5zrr91yi9f2wori=">aaab/xicbvdlssnafj3uv62vajdubluhgpqklnrzdkhlcvybbqit6aqdonkwdyge4qe4ehterr/iyr9x0mahrqcundnnxube4yemcmlz30zpbx1jc6u8xdnz3ds/ma+puijwhjmojlnm+z4shngidcsvjpqttldom9lzpze533skxna4epbpqtwqjsmauiykljyzcut1cg8byrppleec6wffm2tw05odrhk7idvqoo2zx8nrjfviiokzemjgw4l0m8qlxyzmkkmlsilwfi3jqnmihus42xz5gtzvygggmdcvsthxf09kkbqidx3dgsi5ectelv7ndzqmrtymromsjmkljwlfoixhngqcuu6wzkkmchoqd4v4gjjcuudvyvowl29ejv2nav80nxun1rou8iidy3acgsagl6af7kabdaagkxggr+dnedjejhfjy9famoqzkvgd4/mh2k2rza==</latexit> This becomes the fermion two point function when we impose the equations of motion. At low energies, it develops an approximate conformal symmetry.

27 <latexit sha1_base64="lnncgjreqw+tofftbchgzkiyrlg=">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</latexit> <latexit sha1_base64="hw18lxdobfhq7fukondlsy62lrc=">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</latexit> <latexit sha1_base64="j3jwp2nihnb9eytmyb/6mfrcefm=">aaacdnicbvdjsgnbeo2jwxy3qecvjyngjwfmpohfccromyjzibmhnk4nadkz0f0jhel+wjof4klqek/epfk3dpadjj4oelxxrvu9pxzcgwv9g5ml5zxvtey6ubg5tb2t292rqsirlfvpjclz8iligoeschwea8sskcaxro73l8d+/yfjxapwdgyxcwpsdxmhuwja8nkfar72kd7hkr6cz+miem7wpi06uhxfbba8wqocl8tbjwscvejsgcmjgspe7qvvjmgssbcoieo1bssgnyusobvszlysxwjc+6tlmpqgjgdktsffjpcrvtq4e0ldiecj+nsijyfsg8dxnqgbnpr3xuj/xjobzpmb8jbogiv0uqitcawrhked21wycmkgcags61sx7rfjkogatvonym//vehqtsk+ktm3tr58mcsjiw7qitpgnjpfzxsdkqikkhpez+gvvrlpxovxbnxmwzpgbgyf/yhx+qmtfjju</latexit> Scaling solution. G c = t 1 t 2 2 In the IR there is a family of solutions that are obtained by applying a time reparametrization to the above one, u à f(u). G f =[f 0 (u 1 )f 0 (u 2 )] G c (f(u 1 ) f(u 2 )) = h i f 0 (u 1 )f 0 (u 2 ) (f(u 1 ) f(u 2 )) 2 This is only an approximate symmetry and it is explicitly broken. G = G f + G? =[f 0 (u 1 )f 0 (u 2 )] h Kitaev i 1 f 1 f 2 + G 2? (f(u 1 ),f(u 2 ))

28 <latexit sha1_base64="u98gfnhjgihgcgvvf0voxgujygc=">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</latexit> <latexit sha1_base64="etxxgvi9uqtqvosqlmdhf8hdd30=">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</latexit> <latexit sha1_base64="9kacp4ku/gh4ueqlb86oj5/isrk=">aaaclhicbzc7tsmwfiadcivhfmbkswiq2qukyycxkkshhilorwqryhhd1qpzke0gvvheh4lhyqejeglloxdadndys5z+fecchz/fixgv0ri+tmla+sbmvnfb39nd2z8wdo/aiow5ji0cspb3psqiowfpssoz6uacin9jponnr7n654fwqcpgxs4imvdrokajipfuydxq0drhoyf9jbhspc6dful9imcyidb1/swuuhxwzu9uyk7fhzum7holq2rnbvennzssynv0jzf+mmsxtwkjgrkiz1urhcsis4ozsfv+leie8bsnsu/zaknlg2r+awrpfbncucjvcysc098tcfkfmpme6vsrnijlwgb/q/vioboajdsiykkcvfg0ihmuicycg0pkczzspgzcnkq/qjxbhggp4twzfozlm1dn26naf1xn1inv6nkerxactkez2oas1eadneelypainsebeneetfftu/tatba0foyy/jh2/qmw86ob</latexit> <latexit sha1_base64="7yexsb9brj8eofzn2syuoy9bsgq=">aaaccnicbvdlssnafj34rpuvdelmsbeeosr1oruh6mzlc/ybbrom00k7dcyjmxohhg5d+smubavx6x+48m+cpflo64hhhs65lzv3+dgjutn2t7gyura+svnakm/v7o7tmwehbrklapmwjlgkuj6shngqtbrvjhrjqrd3gen4k9vm7zwqiwku3qtptfyorienkezks54jowvb2bwgj8fz2pt4aovv6hptw1bzmyt21c4bl4ltkaoo0pdmr/4wwgknocimsdlz7fi5krkkykzm5x4isyzwbi1it9mqcsldnl9kbk+1morbjpqlfczv3xmp4ljoua87ovjjuehl4n9el1hblzvsme4ucff8uzawqckyxqkhvbcs2fqthaxvf4v4jatcsodxzljwfm9eju1a1bmo1pq1sv2mykmejsejoamouar1cacaoauweatp4bw8gu/gi/fufmxbv4xi5gj8gfh5azzflfi=</latexit> <latexit sha1_base64="yhawqlfdnyes8hoesrqafqkbvvs=">aaacxnicbvfbsxwxfm6m1uq06movqi8p3ykcldp2occc1evposi4kuwssyb7roozzjc8ezdh9+pv8+xjtz37k5rz9acrlws+fo/7emmxpfdsuhg+ev7m7ke5z/mlwzevi0vljzxvm5uxrmbb5co3fwm3qktgnklsefey5fmi8dy5pqr75zdormz1kq0k7gb8ustuck6o6juionmevxbtdrhukq1gg0bjgs04qejqxyir+and2hotg21pzkoqtuducaqvtfvqdedojonu3mlfhzvnxmmjbixjgvcgegebh9hobv9+5vm413ii+7kom9qkfle2e4ufdstusaqfwyaulrzcxpnl7dioeya2w43tgcipx/qhzy3bmmdmvnzupln2kcvomxg6sto9mvyo1ykp3etwuhcloratqwmpwgvurw19avcqgjjahzhuricuuogc3icegushmn7ze3c204p+tnzoxby/2ktm2xe2zjzzxhbzifvnjlmbcfbkmw/bc7x//py/6c9ppl734vng3ps/9h/r3roy</latexit> <latexit sha1_base64="ki6tzmugcpsnk3qi9hywvoagjj4=">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</latexit> Low energy SYK action S = N s J R {fl (u),u} N s J R {fr (u),u} + S conf [? G] {f,u} = f f 0 2 f 002 f 02 X a = 1 f 0 (1, cosh f,sinh f) Same as the action of the boundaries in the gravity theory (H l H m H r )/SL(2) g Q a l + Q a m + Q a r =0 Analogous to the action of matter for the gravity theory. Independent of f(u). Becomes exact in the limit: N!1, ( J )!1, N J =fixed S 0!1 <latexit sha1_base64="em8ucsbqtjtqe3oxbymalb5fqnc=">aaab/3icbvdlssnafj34rpevfd24gwwfvywpc12wunhzon1ae8jkommhtizhzike0iwf4kpqflfu/qnxbvwwp4+fth64cdjnxu69j0gylcq2v4yl5zxvtfxchrm5tb2za+3tt2scckyaogax6arieky5asqqgokkgqaoykqddk/gfvuoceljfquyhhgr6nmauoyulnzrejzk8ma3oati6fieqkwrvlw0y/yecje4m1kshjw+6uvto+5bn24vxmleumimsdl17er5orkkykzgpptkkia8rh3s1zsjiegvn5w/gqda6cewfrq4ghp190soiimzkncdevidoe+nxf+8bqrcsy+npekv4xi6kewz1k+os4a9kghwlnmeyuh1rrapkeby6crmu6fgzp+8sfqvsnnerjr0hduwrqecgxnwbhxwaarggtrbe2cqgwfwbj6ne+predxepq1lxmzmapyb8f4dcewxbq==</latexit>

29 Construction of gauge invariant generators

30 <latexit sha1_base64="dg4ae4e3ssei5lvceyp7nydxquu=">aaacgnicbvdlsgmxfm3uv62vqks3l7zcrsgzdahlohuxldghdmqqsdm2ndmtkowwlp6fupjtxakk4li3/rszbrdapra4nhmvn+f4gjolbxtmzdbwnza3stu5nd29/yp84vflrbektekihsmojxxllkrnztsnhsepdnxo2/74jvxb91qqfov3ohg0f+bhyaamyg0kl28dleoarma8t6eorpcr0bguox5p1yaq6hjydcrq8aiz6+wldswea/4sz0mktyj7/jcrjxuv/+n2ixihnnsey6w6ji10b4klzottac6nfrwyjpgqdg0nstnzm8yttehukh0yrnk8umnc/bkxwyfssecbyqdrkvr1uve/rxvrwvvvwkiraxqsxafbzmfet2ucppouaj4ygolk5q9arlhiok2zuzxpwvnn/je0qhxnoljtmdqu0qjzdiikqiwcdilq6bbvurmr9iie0st6s56sf+vd+limzqzlzjh6bevrg//pn94=</latexit> Momentum generator <latexit sha1_base64="poh8mft7jdh2fn6ovwiqctyqyze=">aaacaxicbvc7sgnbfj2nr7i+vi0sbaytwulcbiy0japigcekc8lmmz1mkigzs8vmrbcwppe7rkwebbg18r+s/anrv8djo9dogquhc+7l3nucmfgpbpvdymznlywuzzfnldw19q1rc6sqo0rguseri4qbieky5asiqglejqvbycbileidj/zadrgsrvxa9wpihajdaztipltkwzsq5vpqbskfwum4hlq+acit+vbolthjwl/emzjckf928fl1d1r2rfdgk8jjsljcdelzd+xyesksimjgbmyjksrguic6pk4prygrxjp+yad3tdkc7ujo4gqo1z8tkqql7ieb7gyr6spzbyt+59ut1t7xusrjrbgoj4vacymqgqm0yiskghxra4kwoppwiltiikx0zqapu3bmf/5lqswcc1qoxuk4zsaewbal9sabcmaxkifluayvgmea3inh8gtcgg/gs/eyac0y05lt8avg6zeuo5gf</latexit> Vectors: Xl a, Xa r Construct gauge invariant generators. First P / (X l X r ).Q m Translation along the geodesics joining the left and right points

31 <latexit sha1_base64="9khdoesh0mgtqocxihqcynkrrh8=">aaacexicbvdlssnafj3uv62vqetfblvrhysklnrzdooybdsg2ham02k7dcyjmxohlhbn0pwf4qacgrh16cpv8cecpgrtpxdhcm693htpedmqlw1/gqmfxaxllfrqzm19y3pl3n6pycgrmjrxxclhbkgsrknsvlqx4sacib4wug06vyo/ekuepff4o7ox8thqhbrjmvja8s1jchm5cf2fqquwfa4hp4mful6yirmcb2ztyx4dzhnnsrkf/whp6+5gwptnj3ojwgknocimsvlz7fh5psquxyz0m/vekhjhdmqrmqyh4kr6vffdfxiklqzsrkjxqoby/t3rq1zklg90j0eqlwe9kfifv0tu88lr0tbofanxzfezyvbfcjqobfbbsgjdtrawvn8kcrsjhjxomjprktizp8+tst5yzqx8scdxcszigz1wce6aa85bavydiigddo7bi3ggl8ad8ws8gm+t1pqxndkff2c8fwnccj22</latexit> Two others: X l.q m, X r.q m Boost Linear combinations give Energy

32 These are expressions for three gauge invariant SL(2) generators, G A = e A a(x l,x r ) Q a m <latexit sha1_base64="zjjv/knbo1alwjsziwzr9tkt8cc=">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</latexit> They act on the physical Hilbert space. à Hilbert space is infinite dimensional Do not confuse them with the SL(2) g constraints. They include quantum gravity corrections (finite Schwarzian coupling). Do not include effects of other topologies (S 0 à ) The X dependence can be viewed as a gravitational dressing.

33 <latexit sha1_base64="k+52d4sjdnbwsuryfgmzmwe/vp8=">aaacf3icbvbls0jbgj1rl7ox1blnkaykjvfaojab2qiwlqzyaxq9zb1hhzz7ygzuibd/rav+rrtwqvg0jvb9jx5anfddlhzg4hdo+fjmo7bpqjc6/qnffhaxllfiq4m19y3nret2tk14acekij3m8yanbghujvvjjsmnnxpk2izu7cfz5ndvcbfuc6/l0cemg3ou7vkmpjks5cge6tq8b5cyek7pwlpyultdhdquwwullzobxxiwshg2clrjlj7xx4dzxjisvdgtf4x33y8vk/nr6ng4cigrmuncna3dl2aiuksykvgifqjiizxapdju1euoewy4pmsed5tsgv2pq+dkofz/t4tiewlo2crpinkxs14k/uc1a9k9mupq+oeklp4s6gymsg9ghceo5qrlnlqeyu7vxyhui46wve0meqofy/bmevir5i2jfofk1vege8tbhtghgwcay1aef6acqgcdw/aansczdqc9aq/a2yqa06yzu+aptpcf6dceag==</latexit> <latexit sha1_base64="yecd2o0hr5jeea3e3ayxotwvgw8=">aaacaxicbvdlssnafj3uv42vqasxboa2qkupsv3osupclxxsa9oyjtnjo3tyygyihncvv+afubiuxk2f4ap/46ttqlspxdiccy/33ungjappmmmtt7s8srqwx9c3nre2d4zdvayiy45ja4cs5g0xccjoqbqsskbaesfidxlpucprzg89ei5ognzljck2j/ob9shgukmocqbhqqthzcnloxbykywdfpxjjle0k+yecjfym1ksfbonz+naunem724vxlfpaokzeqjjmzg0u8qlxyym9g4ssitwepvjr9ea+uty6esbetxssg96ivcvsdhrf0+kybci8v3v6sm5epnejv7ndwlpxdgpdajykgbpf3kxgzkewrqwrznbkiwkimypuhxiaeiis5wzrqsurpmff0mzwrhoktu7fccvmcipdkebliefzken3ii6aaamruafvif37ul71t60z2lrtpvn7im/0l5+aneqlgk=</latexit> They do not commute with the Hamiltonian since X l (u l ), X r (u r ) depend on time. Only depend on time due to the boundary positions. G A (u l,u r ) To the extent that we can solve for the boundary positions à we can write G A (0, 0) = A B GB (u l,u r ) Conserved charges Operator in the boundary theory (Schwarzian)

34 <latexit sha1_base64="wchyu+/egilp+m26vbmclatq5ls=">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</latexit> <latexit sha1_base64="s9mhpxrmwgv9ribbxghmjrtvfqo=">aaacdhicbvdjsgnbeo2jw4zbqecvtrihioazenclebxeywjmgsxdtadjmvqsdpciietuyt/wfzwjcij48gc8+qee/qi7y0ethxq83quiqp4bciavzx0asbn5hcwl+hjizxvtfcpc3crlibkelkjaa1f1qvloffpsthfadqufz+w04nbph37lhgrjav9k9ula8odaz21gqgnjmzmyp9o46hhnwpgeh+bmsqkox/tae03y065jpqysnqkejfaepplpt4uv7/vtgmn+1fsbitzqk8jbypptharrb6ey4xsqqeeshkc6ce1rmvrgudnoj34z4f2ttha7elp8hufq74k+efl2pfd3eqa6ctobiv95tui1jxt95oeroj4zl2phhksad4pblsyoubyncrdb9k2ydeaautq+reknye//pevkuax9mm0vdrxnaiw42kfjlee2okj5dikkqiqiukup6ak9g3fgo/fivi5by8zkzhv9gfh+a6h+mn8=</latexit> Expressions purely in terms of boundary quantities. Use the gauge constraints to write Q a m = (Q a l + Qa r) Rewrite them all in terms of boundary quantities, and their derivatives. P =(@ ur ) log[ 2X l.x r ]=(@ ur )` <latexit sha1_base64="cx5secxtudourakpj6uombomb8i=">aaacchicbzfnsynbeiz7xl0/4upg9yiowg4uehfdtdysf0hci0cfo0ishp5oj2ns+ac7rg1drv5bt/4ht/4ca+kw66of0zy8b71ut3wyamxr8x4dd+bl19m5+yxk4rel5e/vldulm2rgylzidgkuqm6lvrfso0itr1ijerrqerle/yn8yxtprericxynshfxyawgsnakkajeb4n32o0nchebhl+qbhgdpwaaogvucitcmor2ah7c3j/7l2rs69bnuuhfdzbngz4u7hvbtbruqqk6cynmshhqbjrgo6jwvky3lfgifgk1vtzpuh2gysklzixcc2s7vpdily/gcs0nlw5mzcrfnr/kdmhmi2l7+xrfe9ghpq+dxnaxi0zvt4mcr9aoo5a6i44j+94rxm+8toadg16u4jrdgyvxqynmayzqlb/6ykibekzahvf0vxajbrhaeqjkhbbgv//nj3dravr7zdzzq3z0xo5jnq2zn6zofpabhbetdsratlanz9vzdzacz3fn3xs3xltdp8z8yp+vu/scvpw1tg==</latexit> 0=ẋ l + p m +ẋ r! p m = ẋ r ẋ l =(@ ur )(x r (u r ) x l (u l )) Similar expression for the other two. It involves the distance between the two boundaries.

35 <latexit sha1_base64="iyiwmifno2s7yq7dehned7u99ng=">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</latexit> <latexit sha1_base64="yhawqlfdnyes8hoesrqafqkbvvs=">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</latexit> <latexit sha1_base64="ccp/1kjmycgn1zei+wjzwgalqq0=">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</latexit> Distance between the two boundaries 1 ( 2X l.x r ) /h j l j ri Could be extracted by taking the logarithm of the correlator. ` = log[ 2X l.x r ]= 1 log[ j l. r] j Well defined as long as the operator does not vanish. Is OK in the scaling limit we defined before. N!1, ( J )!1, N J =fixed It is a quantity that is well defined only around this wormhole phase. Like the phase of a superconductor. Previously thought of in terms of complexity Susskind et al

36 Now we will now work to get more explicit, but approximate, forms for the generators. These will have the advantage of being well defined in the finite N theory.

37 <latexit sha1_base64="2q/gb/u0g3d3dgs/awvwqbnqwng=">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</latexit> <latexit sha1_base64="o79azidusynw4nqnqccephtkncc=">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</latexit> Semiclassical limit S S 0 r3 e T G N N J 1 Gravity (Schwarzian) is weakly coupled and close to classical. Boundaries follow classical trajectories + small fluctuations. AdS 2 matter t l =ũ l + l (ũ l ), t r =ũ r + r (ũ r ), ũ 2 u

38 <latexit sha1_base64="zn7aseqawskgkwvzrjqq9ba7tda=">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</latexit> We can now calculate the distance in terms of classical solutions + small fluctuations. The generators G A have a simple approximation in terms of ε B 0 r 000 r ( 0 l 000 P 00 r 00 l Ẽ ( 000 r 000 l ) l ) All evaluated at u l =u r =0

39 <latexit sha1_base64="eg2npldsutjuxrtnjjcx2xqdzva=">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</latexit> <latexit sha1_base64="2q/gb/u0g3d3dgs/awvwqbnqwng=">aaacgnicbvfnaxsxenvu3cr1m8rtjr0mtqvo17lrhfpocyg55jha7qs8zthk40re+1fpnmaw+9q/kl/uc0/9c/0rjbidasfua8htm/fqacbolttk+78dd63yyn1j82x11eut7z3am7cdkxvayftkktpxmteoziptkqtwotfik1jhvxx3nqlf3am2mku/0tdhxsjvujmqgpovotodqkmbqjgcrxcsvh2ewl4oimtcsgu9qjnx9wdgh+mzknklgb2y0fom3pth/rmt83sh76gefos5naizbrpcgindypyu1eq+508bqyr4ivwtypzj88+1pxdr7vfyz0sryepccwo6gz9tr+sapfa4qoafwzylo36dxuttnqdpldmjjucdvfowylq9kcfuxuyupdfmmmtwmxc6ncu1ifi/wregwadekdo8iezf7kfboyaymkwd+lkjidw0hastba8gbrnmguzsqlu7hwd5z6uk0/kcy691acdxymbyzo/ye9zkafvittg5u2btjthfp+ecoz5bcffdwd2ewv3nkbplnsh98gi3ychx</latexit> <latexit sha1_base64="7yexsb9brj8eofzn2syuoy9bsgq=">aaaccnicbvdlssnafj34rpuvdelmsbeeosr1oruh6mzlc/ybbrom00k7dcyjmxohhg5d+smubavx6x+48m+cpflo64hhhs65lzv3+dgjutn2t7gyura+svnakm/v7o7tmwehbrklapmwjlgkuj6shngqtbrvjhrjqrd3gen4k9vm7zwqiwku3qtptfyorienkezks54jowvb2bwgj8fz2pt4aovv6hptw1bzmyt21c4bl4ltkaoo0pdmr/4wwgknocimsdlz7fi5krkkykzm5x4isyzwbi1it9mqcsldnl9kbk+1morbjpqlfczv3xmp4ljoua87ovjjuehl4n9el1hblzvsme4ucff8uzawqckyxqkhvbcs2fqthaxvf4v4jatcsodxzljwfm9eju1a1bmo1pq1sv2mykmejsejoamouar1cacaoauweatp4bw8gu/gi/fufmxbv4xi5gj8gfh5azzflfi=</latexit> We can also expand correlators in ε and, as long as we get the same combination à we are getting these generators. apple t 0 l t0 r cosh 2 ( t l tr 2 ) t l =ũ l + l (ũ l ), t r =ũ r + r (ũ r ), ũ 2 u The gauge constraints set these combinations of ε equal to the matter charges. (In the appropriate gauge). Q a l + Q a m + Q a r =0

40 <latexit sha1_base64="/jwgw8ruaivrkaon+vnzcxkth4u=">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</latexit> Global energy Two identical coupled SYK models JM, Qi H coupled = H r + H r + iµ j l r j <latexit sha1_base64="wkggbeuk2cp2xvbzm6dqlwuxib0=">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</latexit> Ground state is very close to the thermofield double state at some temperature given by a combination of μ and J. Ẽ 2 [H coupled hh coupled i 0 ] β dependent through μ. Redshift factor, makes it equal to unit normalized global AdS 2 energy Empty wormhole

41 <latexit sha1_base64="4qyaoy7rtws8f+8w/ws0slxosl8=">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</latexit> Ẽ ( 000 r 000 l ) H coupled hh coupled i 0

42 H coupled = H r + H r + iµ j l r j <latexit sha1_base64="wkggbeuk2cp2xvbzm6dqlwuxib0=">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</latexit> Involves a coupling between the left and right systems Related to the traversable wormhole construction Gao, Jafferis, Wall

43 Side Comment Rindler coordinates. Global energy is Similar to: H coupled = H r + H r + iµ j l r j <latexit sha1_base64="wkggbeuk2cp2xvbzm6dqlwuxib0=">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</latexit>

44 End of side comment

45 Operator-State map in NCFT 1 Operators are mapped to states of the two sided system. Vacuum à TFD. Matrix elements of correlators à Related to OTOC

46 Related to out of time order correlators or OTOC Measuring distance via correlators Bulk matter OTOC à simple bulk displacements. Here we simply invert the logic.

47 Relation to size Some papers have been proposing a relation between the momentum of bulk particles to size. It was proposed as a conjecture, or ``experimental relationship, with some rescaling factors, etc. Roberts, Stanford, Streicher; Susskind; Brown, Gharibyan, Streicher, Susskind, Thorlacius, Zhao; Qi, Streicher Size is the number of operators that we have to apply to the infinite temperature TFD to create the state we care about. The infinite temperature TFD is a very simple state consisting of essentially Bell pairs of fermions Size can be measured by the left-right correlator. Ŝ = i <latexit sha1_base64="lk9j7chk+vafa1jbmk0arkhxae8=">aaaccnicbvdlssnafj3uv42vqetfblvbvunqqjdc0y3lfu0dmham00k7dpjgziku0kwu/brxgljx6x+48hv8csenc209clmhc+5l5h4vzlri0/zucguls8srxvv9bx1jc8vy3mmjkogynhheit7xkccmhqqpqwske3ocao+rtje8zpz2legcrugnhmxecva/pd7fscrjnsasl6e9qbjew3nior0l6rk88cx0jzjzmaea88t6iaxa/qtxdxcwqbvgh92lcbkqugkghohaziydfhfjmsnj3u4eireeoj7pkhqigagnnv4yhkdk6ue/4qpccafq740ubukmak9nbkgoxkyxif953ut6z05kwzirjmt5q37coixgfgvsuu6wzcnfeozu/rxiaeiisxwerqsurnmb50mrwrfoktwgiumc5cicpxaijoeftkenxie6aaim7sejeayv2op2pl1qb/loqfvz2qv/ol1/axajmtq=</latexit> l r H coupled = H r + H r + iµ j l r j <latexit sha1_base64="wkggbeuk2cp2xvbzm6dqlwuxib0=">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</latexit> Gives a precise relation

48 <latexit sha1_base64="n/urxuln1ejs1g24i7zdinz2wfq=">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</latexit> Bulk momentum is related to the time derivative of size P / [B,E] / [H r H r,h r + H l + µs] / Ṡ This relates ``size to symmetry generators and ``explains its previously observed properties.

49 Order from Chaos H coupled = H r + H r + iµ j l r j <latexit sha1_base64="wkggbeuk2cp2xvbzm6dqlwuxib0=">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</latexit> Only this term grows

50 Comments about exploring the interior

51 Interactions behind the horizon

52 H = H L + H R Evolve the TFD, backwards, insert some excitations with unitary operators.

53 We expect that the initial state is describing the Wheeler de Wit patch

54 We can evolve it with the decoupled hamiltonian H = H L + H R From the boundary theory they should NOT Interact in any way. Fortunately, their interaction is behind the horizon so we cannot see it from either boundary.

55 We can evolve it with the coupled hamiltonian H = H L + H R + H int We can now see the interaction. It is OK because the underlying Hamiltonian has an interaction between the two sides. Make that interaction behind the horizon more real. But always through the lens of a particular evolution. create Acting with two sided operators we can see the interior

56 A black hole is not a ``state. It is a state + a particular time evolution. It is a space-time after all

57 Bulk near inner horizon? 1) We do not know the boundary conditions for the bulk matter beyond the region covered by the physical boundaries. 2) Similar to Coleman de Luccia decays to AdS. We expect that corrections due to irrelevant operators give a divergence at the inner horizon.

58 Some more conceptual comments

59 <latexit sha1_base64="xapy4xz/dcpop9j7s/dtdvoxlgk=">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</latexit> Analogy between two coupled SYK models and ``superconductors. ``Superconductor = System with a spontaneously broken U(1) global symmetry. SYK model for charged fermions. H = P ij;kl i j k l Sachdev Two copies plus boundary interaction. (q >4 is simpler) H coupled = H l + H r + l i i j r l r j <latexit sha1_base64="clc9owveymtiqpnf8dxw1vmcejk=">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</latexit> Preserves U(1) l xu(1) r

60 <latexit sha1_base64="7nmeywlyhxbsnpe/3sc4/bpg8a4=">aaacahicbvc7tsmwfhxkq4rxaimfxaifmvvjgwbbkrb0lbj9se1uhndprtpozduvqtcfma9gqgijsfiljex8bz+a+xig5uiwjs65v/f4+dgjutn2l5fzwfxaxsmummvrg5tb1vzotuajwkskixajho8kyzstqqkkkuyscap9rup+72rk1/tesbrxgzwiireidqcbxuhpqwxt5fpq7smrdyk8hy7lcl7afl5l5eycpqacj86u5ephbvnh9uo70ri+3xaek5bwhrmssunysfjsjbtfjaxnn5ekrrihoqspkuchkv46zj+eh1ppwyas+ukay/x3ropckqehryddplpy1huj/3nnravnxkp5ncjc8erqkdcoijgqa7apifixgsyic6qzqtxfamglkznn3yiz++d5uiswnjnc8vrxcqkmyij9cacogqnoqqmuqqvuaqz34be8gxfj3ngyxo23ywjgmo7sgj8w3n8a7huxhg==</latexit> <latexit sha1_base64="7nmeywlyhxbsnpe/3sc4/bpg8a4=">aaacahicbvc7tsmwfhxkq4rxaimfxaifmvvjgwbbkrb0lbj9se1uhndprtpozduvqtcfma9gqgijsfiljex8bz+a+xig5uiwjs65v/f4+dgjutn2l5fzwfxaxsmummvrg5tb1vzotuajwkskixajho8kyzstqqkkkuyscap9rup+72rk1/tesbrxgzwiireidqcbxuhpqwxt5fpq7smrdyk8hy7lcl7afl5l5eycpqacj86u5ephbvnh9uo70ri+3xaek5bwhrmssunysfjsjbtfjaxnn5ekrrihoqspkuchkv46zj+eh1ppwyas+ukay/x3ropckqehryddplpy1huj/3nnravnxkp5ncjc8erqkdcoijgqa7apifixgsyic6qzqtxfamglkznn3yiz++d5uiswnjnc8vrxcqkmyij9cacogqnoqqmuqqvuaqz34be8gxfj3ngyxo23ywjgmo7sgj8w3n8a7huxhg==</latexit> <latexit sha1_base64="7nmeywlyhxbsnpe/3sc4/bpg8a4=">aaacahicbvc7tsmwfhxkq4rxaimfxaifmvvjgwbbkrb0lbj9se1uhndprtpozduvqtcfma9gqgijsfiljex8bz+a+xig5uiwjs65v/f4+dgjutn2l5fzwfxaxsmummvrg5tb1vzotuajwkskixajho8kyzstqqkkkuyscap9rup+72rk1/tesbrxgzwiireidqcbxuhpqwxt5fpq7smrdyk8hy7lcl7afl5l5eycpqacj86u5ephbvnh9uo70ri+3xaek5bwhrmssunysfjsjbtfjaxnn5ekrrihoqspkuchkv46zj+eh1ppwyas+ukay/x3ropckqehryddplpy1huj/3nnravnxkp5ncjc8erqkdcoijgqa7apifixgsyic6qzqtxfamglkznn3yiz++d5uiswnjnc8vrxcqkmyij9cacogqnoqqmuqqvuaqz34be8gxfj3ngyxo23ywjgmo7sgj8w3n8a7huxhg==</latexit> Wormhole solution or TFD à breaks spontaneously U(1) l U(1) r. But preserves U(1) l + U(1) r Goldstone mode ' = R A AdS 2 ' = R A If instead we had the interaction: H l + H r + iµ( j l j r + j r) l j <latexit sha1_base64="yrjexwh2vbavp3rzpaaqjpgar+o=">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</latexit> Symmetry would also be explicitly broken, and one value of would be selected. ' = R A State displays spontaneous + explicit breaking of this symmetry.

61 <latexit sha1_base64="qn6/t3kgftyhlirs2yhmmt3qfde=">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</latexit> Time superfluids We start with two time translation symmetries: H l and H r The wormhole, or the TFD state break spontaneously H l + H r. But preserves H l - H r The Goldstone mode is the relative time shift between the two sides. This is one of the physical modes of the wormhole (the other is the mass of both black holes). ui = e ih l u TFDi/ P n e i ue n E n /2 Ēni E n i

62 <latexit sha1_base64="mekzww+wsnr21ysqr7e8zp7tc+4=">aaacbnicbvc7sgnbfj2nr7i+vgublqazgice3vhoi4royzmaeucyllotstjkdnazmrxcksbkt7frubbbc//ayszvcfionphavrzouzeze4kyuakc58vilcwula9kv8219y3nlwt7pyajrgbsxrglrcnakjdksvvrxugjfgsfasp1oh858uu3rega8rs1iikxoi6nhyqr0pjvhdg2bf0rphbm4avmfafpdgcqqtv2rzytd8aa88sdklxxr/jnh0sfzd/6blujnisek8yqle3xizwxiqeozmrothjjyot7qeuamniueuml4yug8egrbdijhc6u4fj9vzgiumpbgojjekmenpvg4n9em1gdcy+lpe4u4xjyucdhuevwfalsu0gwygnnebzu/xxihhiikx2caeou3nmb50mtkhdp84wkjqmejsicfxaijoelzkarximyqaim7sadeayvxr3xzlwab5prjdhd2qv/ylz/aiojmm0=</latexit> Time shifted one. AdS 2 AdS 2 Ordinary evolution is just a simple motion in the space of time shifted wormhole: u = u r + u l Goldstone is linear in time because the Hamiltonian is the broken symmetry.

63 Space-time superfluids The time-superfluid picture is valid for any wormhole or TFD. When the wormhole or TFD are those of a Nearly AdS 2 or CFT 1 Then on each side we have more symmetries, including and approximate SL(2) conformal symmetry. The TFD is breaking this to a common SL(2) symmetry. The Goldstones are: The mass and the time shift. In addition the symmetries are explicitly broken by the Hamiltonian.

64 Conclusions We explored the symmetries of the near horizon region of near extremal black holes or SYK modes. Constructed ``exact SL(2) generators. Discussed approximate expression in the semiclassical limit. Explained why and how ``size is connected to energy and momentum.

65 Slogans Order from chaos. Time superfluids Spacetime superfluids.