Exploring Landscape, renormgroup quantization. Maxim Budaev Abstract

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1 Explorng Landscap, rnormgroup quanzaon Maxm Budav 8 Absrac In hs papr h Landscap ponal s consdrd as an nvronmn for sysm: rajcory-nvronmn (E) h rajcory s gnrang a masur on h landscap h nropy of hs dynamc masur s a powr facor for rajcory hs dynamcs lads o a mmory apparanc and producs a non-sngular masur praccally ndpndnly from nal condons I s shown ha masur s dual o h mrcs and s voluon may b consdrd as h landscap dformaons: producon-dsrucon of aracors-vacua I sms lk h rnormalzaon procss wh phas ransons: rnormgroup quanzaon I s shown wha h nropy of hs global landscap masur acs on h rajcory alk a dark nrgy Emal: maxbudav@yahoocom

2 Inroducon h Landscap paradgm [][] dmands a dynamcs for slf xplorng hs dynamcs mus dfn h lfms for h masabl vacua and a non-causal nracon bwn hm Dscr, dns spcrum of a landscap ponal rqurs a dynamcs, whch has a jump rgm alrnang by a prurbav on For hs purposs, on usually uss an nsmbl: h nsmbl of nal condons [7][8][9][][][] In hs papr w shall consdr a consrucv approach o hs purposs Namly, w shall consdr h rajcory of xploraory procss as a squnc of masurmnsdcohrnss, whch producs h confguraon spac n a form of masur W shall s wha hs dynamc masur s dual o h gomry of landscap ponal: o a mrcs n Kählr sns In consdrd modl (E) h nropy of hs masur s a ral conformal dynamc facor for a mrcs on confguraon spac hs dynamcs lads o a mmory apparanc h mmory s rmarkably mrgs on larg scal (on scal of aracors (vacua) dynamcs) as a opologcal (non-prurbav) convrgnc o a global ponal s flanss hs convrgnc dos no lad o saonary hr ar nabld h larg flucuaons whch uplfs h mag of rajcory n h landscap nrgy spcrum Nx rlaxaon s a squnc of masabl unvrss wh ndncy o suprsymmry, and so on In h rnormalzaon doman, h masur producs an ffcv (rnormalzd) nvronmn W shall s ha h dynamcs on landscap spcrum may b consdrd as a rnormgroup quanzaon h papr s organzd as follows - 7 parly rvw of [3] In w dfn h aracor (masabl vn) and characrz n rms convnn for followng smaons In 3 w prov h bfurcaon rlaon (dualy condon) h mporan rsul for local dynamcs In 4 w sma h lfm of aracor and hnc show wha aracor s no xacly aracor; hs objc has svral phass of voluon ncludng a dsrucon phas Durng h lfm smaon w shall s hs phass n dal W dfn mporan valu-h nropy producon-h bfurcaon producon In 5 h scal of aracors dynamcs s consdrd W shall s ha squnc of aracors lfms s corrlad by mans of mmory In 6 w shall consdr h nropy voluon n mor dal and sma a flucuaon m and a rlaxaon m In 7 h cycls n a squnc of masabl vacua ar consdrd In 8 w consdr h phas of aracor s dsrucon n h dark nrgy conx 9 s dvod o dscusson

3 Dynamcs h gradn flow of a landscap ponal has a vas s of h crcal submanfolds of varous dmnsons and gomry: h aracors-vacua ha draws h rajcors Havng addd a small sochasc rm o gradn-lk sysm, s possbl o absrac from unsabl, crcal submanfolds As a rsul h sysm of h Langvn yp s apparng: D j =-l() ju() j D +Dh Î -dscr m: D =, j à M j() - culmnang pon of rajcory - confguraon spac-oucom spac, In fuur, for smplcy, w shall nam { } rajcory j and () j by h U ŒC ( M) - nvronmn-landscap ponal, h( ), h () = - sochasc procss-quanum flucuaons () W shall assum, for smplcy, h nos s unform dsrbud n a small boundd - hh, h nos s mporan, frs of all, as a dsrucon mchansm for nrval ( ) unsabl, crcal submanfolds of gradn flow l = l() Œ -bfurcaon (rnormalzaon) paramr-powr facor In fac, s h mrc facor l º l() dj, s dfnd by mans of masur For dfnon of h masur s gnrad by a dscr rajcory mbddd n a confguraon spac, ha may hav arbrary dmnson, w consdr som paron x of h confguraon spac on vns (coars-gran) W assum h avrag damr of paron s clls much mor hn nos ampludh L s dfn h dynamc frquncy masur m on hs paron: L a h m h rajcory has spn h m- n ( ) n a paron s cll x hn w dfn: n ( ) m( ) Obvously, Hr V ( )  m( ) = for all V Œ à x -s of vsd clls a h m V hus, m Œ Sym( ) 3

4 Now dfnl Dfnon Enropy of ndvdual rajcory s:  V Œ h () h(,) x =- m() log m() Formally, s h nropy of masur m on paronx L V A las, w pu by dfnon: V ( ) Œ - s capacy of vsd clls s ( ) l( ) log V h ( ) l - () Hr l -posv consan, dmnsonal nvrs mass No, ha h bfurcaonal paramr l s a nonncrasng funcon of formal nropy l can b consdrd also as a dynamc varabl, masurng a dvaon of h currn dsrbuon from unform on In hs sns h bfurcaonal paramr s an nformaon facor On h ohr hand, s a momnum scal ha s a powr facor In h dual, l -funconal s a conformal facor for mrcs, whch dpnds on h rajcory s "mass dsrbuon n confguraon spac () hus, h dnsy s flucuaons ar xpand/comprss h confguraon spac In hs sns l s a gomry facor (curvaur) 3 Bfurcaonal rlaon h bfurcaonal rlaon (dualy condon) s an nqualy sablshng communcaon bwn h nformaon and h local gomry (nrgy) of a landscap ponal durng h momn of ranson bwn aracors-vacua Frs of all, w characrz aracors n rms convnn for furhr smaons For hs purpos w shall dfn aracon ara low-nrgy lm Frs, w consdr h lmnary aracor-local mnmum In som vcny O of a local mnmum h landscap ponal can b prsnd n a quadrac form wh som accuracy Afr rducon o man axs (propr bass) w oban: n U( j) = U( j) +  w ( j - j) + o(( j-j)) = Hrj-j -local coordnas corrspondng o vacuumj hs coordnas corrsponds o fundamnal flds of a local Lagrang hory w w - local nvronmn s curvaur nsor ha ncodng h symmry brakng hrarchy ( w - marx of couplngs and masss of local quadrac hory Hr w shall nam hs h modul spac) h qualy sgn corrsponds o possbl dgnraon n som drcons n cas of a non-rval local mnmum (crcal submanfold) L's mphasz wha h dffrn local mnma can hav varous codmnsons S [3] for mor dal j 4

5 Rmark h culmnang pon of h rajcory n vcny of a crcal submanfold may b consdrd as a Fynman s prmv r dagram (worldsh) hn h rajcory s an ngral procss of glung h acual dagram o a horzon of mulvrs sa wh bulk (hsory) dpndn couplngs (ffcv nvronmn) In h furhr, for smaons, w shall characrz h aracor by scalar: N w = v =  N - "local avrag curvaur" or "nrgy" of aracor N - aracor s codmnson In smaons w shall us h paron assocad wh a s of all aracorsa: " aœ A, $ xa Œ x: Oa () à xa, and A x -s a bunqu corrspondnc I s possbl o ll, wha h cll corrsponds o h modul L h nrgy of aracor, whch conans n cll x, sv W shall nam s: { v } h powr spcrum of landscap hus w dnfy h paron and s of aracors and characrz hm by avrag curvaurs: { A } { x } { v } Furhr h symbol w wll dsgna vn, aracor or aracor s nrgy (curvaur), dpndng on conx Proposon h bfurcaon rlaon (dualy condon) s: l w > (3) Proof: h qualav pcur of sysm s bhavor n a vcny of a local mnmum s such: Afr h rajcory gs n basn of aracon a h m : j( ) ŒO( j), j Œ A, quckly rolls down on a boom of hol: jæj, : O() hn h rajcory oscllas n a vcny of a boom wh amplud ordr of Dj ; l() v h Furhr, l - funcon vars n complcad mannr unl s magnud dos no bcom suffcn for prformanc of condon: or l( ) v j- j > j- j, l() v - > Dffrnly, h drf facor should chang (ncras) up o such magnud ha an ach nx raon movs away a rajcory from "boom" of a ponal hol 5

6 Rmark h fac, wha l -funcon grows n hs cas, wll b dscussd n dal a smaon of h aracor s lfm I s also vsbl from qualav pcur: localzaon of h rajcory n aracor (masur concnraon ffc) s accompanyng by nropy dcrasng (ncrasng of condonal nropy) and by ncrasng ofl - funcon () Afr prformanc of an nqualy (3) h rajcory vry fas lavs h aracor "Vry fas" mans: log () d() d :, -scal of h rajcory dvaon from a boom of ponal hol Hr - local m Rally d : l() d, and, as wll b vsbl (a curv no black rcangl on Fgur ), locally: l ª cons >, l w - - : W s, ha h rajcory s blow-up unnlng (suprnflaon) hrough a ponal barrr I sas a ncssary smaon and prms fnsh h proposon proof h rajcory n spac of ponals: w() l l() w (4) - s rqurd for us W shall nam hs by ffcv curvaur of ponal I s an llusraon of a rajcory-landscap dualy In h rnormalzaon sns hs dfns a rnormgroup ngral for ffcv couplngs and masss hs s analogous o h ffcv cnral charg n h conformal fld hory If h spac of ffcv couplngs s chosn as a landscap ponal curvaur Som mporan rmarks s appropra mnon hr h blow-up nflaon s allowng us o hav no nrs n h srucur of a ponal barrr (wdh, hgh, c) W shall assum ha h rajcory hav an nsan jump no nx vacuum afr prformanc h bfurcaon rlaon (3) If h rajcory gs n som aracon ara (vacuum) and h bfurcaon rlaon (3) s aand, h aracor s lfm wll b nsgnfcan: : I s possbl o ll, ha h rajcory sll "dos no noc" hs aracor Or, n psychophyscal rmnology, h subjc (rajcory) sll rmmbrs such or a mor srong" gomry, and dos no h bg anon (nrs) for acual vn ak no accoun hs rasons and ha h ndvdual rajcory nsad of nsmbl s consdrd, n h furhr w shall us concp of vn nsad of a sa Hr s prnn o spcfy h concp of masably n our conx Dfnon h vn s calld masabl or sngular on condon ha: v() l, (5) and cohrn ohrws h masabl vns quanavly dffr from cohrn hy by much grar m of lf h concp cohrn s accpd n h lnar dynamcs In our approach can b jusfd by followng qualav rasons: n cas of masabl vn h rajcory scans on aracor and n l -funconal h condonal nformaon on 6

7 aracor s gomry w s accumulad In h cohrn cas, for arbrary boundd "masurmn m" D (h m rsoluon s mor han un), n hs masurmn wll ak par a suprposon of n ªD aracors-vns Dffrnly, n h masabl cas h masurmn s a avragng on h oucoms blongng o sngl vn By conras, n h cohrn cas h avragng occurs, n fac, on s of oucoms-vns In hs dfnon s mporan ha h sam vn-aracor, n du cours, can urn from masabl o cohrn and \or on h conrary Our chncal dnfcaon aracor-vn dos no play a sgnfcan rol hr hs dfnon formalzs a chang of dynamc mods, and wll b dscussd n dal n h furhr 3 h bfurcaon rlaon xprsss h dualy bwn h nvronmn and h rajcory n rms v andm hs crcumsanc s a rflcon of h nformaonal characr of h nracon bwn hm 4 In h furhr w shall no consdr h unnrsng, saonary suaons: l logv v - <, s corrspondng o dla-lk dnsy a lm Æ From a such saonary s possbl o g rd, havng usd mor fn paron or, spakng languag of xprmnal physcs, havng ncrasd rsoluon of h masurmn ools No, ha h rval and ponws parons ar rsul n h purly random dynamcs 4 Esmaon of vn s lfm Dfnon 3 L a h momn h rajcory falls no a basn of aracon of som ponal hol, and has lf hm a h momn + W shall nam by lfm of h corrspondng masabl vn-aracor In hs sns, h frquncy masur of aracor-vn s: R ( ) - r r= m( ) = Â ( ), R ( )-numbr of rurns of h rajcory n h cll x a h m -m of r s rurn hs s h rlav, oal lfm of h vn I s mporan, ha = ( ) s m-dpndn hs s an llusraon of h m hrogny For us s ncssary, by vru of (3), o sma h m of voluon of h bfurcaon paramr o magnud v - W shall assum ha aracor s conand n sngl cll: z L s dfn small paramr: = =, -m of z occurrnc, -currn m of hs vn xsnc (6) hus, global m s + Hr hr ar ssnally varous wo cass: ) h rajcory coms back n consdrd aracor-h rncarnaons cycl 7 r

8 ) h rajcory gs n an "unknown vacuum, ncrasng a volum n dfnon () L's consdr hs cass sparaly W shall nam h vn For small voluon s had: Oban: x acual, f j( + ) Œ x n ( ) + m( ) + m( + ) = = + + n ( ) m( ) m( + ) = = + + -for acual vn, -ohrws Ê V - m + Êm + ˆ m Ê m ˆˆ l( + ) = l logv + log + log Á Á Ë Â Ë + + = + ËÁ+ l( + ) > v - I s asy o s, wha solvng of las nqualy s ranscndnally complx hrfor, w shall rcv squar-law smaon For hs purpos w shall xpand l n powr srs on small paramr up o h scond ordr Ncssy of h scond ordr wll b provd a calculaons I s also vsbl from Fgur and Fgur W hav: l( + ) = l(, ) = l( ) + l + l + o( ) Afr rval calculaons w oban: ( h ( ) log ) l = l + m l l = l + ( m - - ) (7) ) In cas of ncrasng volumv Æ V + s oban: - l( + ) = l( ) + l log( + V ) + l + l + o( ) Hr: - m ª, l l ª l ( h ( )- log ), l ª l + for grar In Fgur 3 s vsbl (smoohd), ha afr occurrnc of an arlr unknown vn hr s a "srss"-splash of nformaon: dl l - : causd by opologcal proprs of an nvronmn V Howvr, hn h nropy vry quckly ncrass: - h : - log, ª, ladng o local rlaxaon (Fgur 3) hs srssful suaon has wo scnaros, dpndng on sgn of quany: l dl v l l log( ) v = + + V and on "hcknss" of aracor s walls hs s or a panc flgh (h rd schdul), or a frgh, rplacd by an nns nrs (h grn schdul) I s wll vsbl n Fgur 3 In h furhr, for 8

9 smaons, w shall us h scond scnaro Hr w shall no sma accuracy of approach W ar nrsd only n ordr of valus and n qualav pcur W nam h valu l by a condonal, normalzd spd of chang of nropy Apparnly, hs quany conncs global h and local log m nformaon Sgn of hs spd can b boh posv, and ngav I xplans wo scnaros of bhavor of h bfurcaon paramr ar rprsnd n Fgur and Fgur For h lfm w oban a rough sma n lnar approach: - - v -l() v -l() > = l l ( h ( ) + log m ) For squar-law approach s ncssary o fnd h soluon of nqualy: W oban: l() l l v > - ( ) 9 (8) - l + l -4l l-v > (9) l L s consdr now wo conscuv bfurcaons: frs, h rajcory falls no vacuumv, hn, afr a whl (lfm), havng xprncd bfurcaon, no a vacuum v ( v > v-dffrnly h rajcory wll no obsrv v by vru of h bfurcaon rlaon (3)) As shown abov s possbl o nglc a m of ranson bwn aracors by vru of smallnss n comparson wh h lfm W shall accp also h >- log m, as s possbl o us lnar approxmaon for ( Fgur ), hn: Or Ê ˆ - Á Ëv v l logv v l log m Dw w l w m log - ( V ) () D w = w for dns spcrum: hs smaons look lk a uncrany rlaon (non-sandard, rnormalzd) I s possbl o nrpr v as a facor of nrgy Rally, locally n m (spac) for an gnvalu of h voluon opraor s oban: j ª E j, E = E() = l() v l ª cons and E v For small m nrvals- : I jusfs our nrpraon hus, h lfm can b characrzd n rms of "nrgy jumps": = ( v Æ v, ) = ( v, v, ) = ( ) j j j 5 Suprmodul spac "Slow m" Abov, a local pcur of h rajcory-nvronmn nracon was consdrd Now, w shall pass o a mor global pon of vw L's consdr h nvronmn as a s of conncd aracors-vacua For hs purpos w dfn a quon paron I s asy o s ha hs s a opologcal, ornd graph:

10 G= ( VE, ) Hr w shall nam hs graph h suprmodul spac alk [] h aracors ar corrsponds o vrcs vrxsv, o dgs E -possbl ransons bwn aracors Each vrx s characrzd by ffcv curvaur (nrgy) of corrspondng aracorvacuum: W : V Æ { v} Hr { v} -powr spcrum of h nvronmn ponal-h s of aracors nrgs (avrag curvaurs for our smaons) wo vrcs vrx ar conncd by h dg: ŒE à V V, j f a drc ranson bwn corrspondng aracors s possbl n h mchansm of bfurcaon (3) h drcon of an dg s dfnd by a drcon of a rajcory ranson For ach vrx v Œ V w shall dsgna a s of ougong dgs as ouv () o ach dg from hs s w shall sablsh a corrspondnc wh a ranson probably bwn h vrcs-aracors: p: E Æ (,),  pj = jœou () hs randomnss s causd by h nos of modl-h In fuur, for smplcy, w shall consdr h lac varan: p = p = dmm -, f j, -ar ndxs of nghbourng clls, and ( ) j j p = ohrws j Acually, a dsrbuon of ransv probabls s gnrally non-unform and dpnds on local gomry of h nvronmn ponal graph (curvaur nsor) For smplcy of furhr smaons w absrac from hs nformaon and accp sphrcally symmrc ponal hols as aracors h vacua lfms gv a naural paron of h m hs paron s no homognous, wha s dsplay a non-saonary of our sysm W ak h facor of hs paron and nam hs facor-m Q as "slow m" In h spaal paron, accpd by us (), s a squnc s ndxng h ransons bwn vns-aracors hus, w oban Markovan procss on h graphg h spao-mporal facor Dfnon 4 W shall nam rpl: F ( GP,, ) () h facorsysm By analogy o h hory of rnwal procsss s h oprang (subordna) Markovan procss W hav h facor-dynamcs: Æ x Æ x Æ Æ n o x Æ () and corrspondng codynamcs: Q Q+ Q Æ v Æ v Æ Æ v Æ (3) j j j Q Q+ Q+ n n h powr spcrum of landscap Lfm dynamcs: (4) k k k Q Q+ Q+ n

11 Bfurcaon rlaon (3) gvs a qualy o aracors (vns) hrfor s possbl o dfn h dynamcs of qualy : Æ q Æ q Æ Æ q Æ (5) k k k Q Q+ Q+ n Symbol q k rflcs a masabl qualy of aracor and has on of wo valus: masabl (sngular), cohrn 6 Flucuaon, rlaxaon I s almos obvous, ha h l -facor s a nonmonoonous funcon of m ha lads o dsrucon and occurrnc of h aracors (5) By way of llusraon w shall sma h m of flucuaon and h m of rlaxaon and compar hm L s consdr followng squnc: F hæ( h -Dh) Æ R h, F -m of flucuaon, R -m of rlaxaon (6) h nropy flucuaon corrsponds o flucuaon of dnsy n h bfurcaon mchansm hrfor, as F s possbl o accp h smaon for lfm of masabl vacuum (8) I s h voluon of condonal nropy Provdd ha h rajcory s localzd n som paron s cll ( s rappd by masabl vacuum) hs dynamc mod w shall nam h flucuaon mod h rlaxaon, n oppos, s an uncondonal nropy growh Such suaon can ars, for xampl, n compac cas: V = cons afr a nx "mnmal bfurcaon, whn h rlaon (3) s ru and curvaur-nrgy of acual aracor s mnmal A long m afr bfurcaon (rlaxaon m) h rajcory dos no noc ohrs vacua and, hus, s a squnc of cohrn vns hus, h sysm dgnras n an ordnary Markovan procss on graph G, whch acually concds wh h oprang procss () L s consdr a compac cas V vss ach cll of paron on avrag = cons W assum, ha n m R V - R h rajcory -ms I corrsponds o a unform, saonary dsrbuon of h oprang procss () hs s ha w nam pr-rgodc consdraon In hs cas h nropy as a funcon of m has smaon: h ( + + ) ª h ( + ) + h, F F h =- ( h + log m), (7) log m = Âlog m( + F) -pr-rgodc assumpon, V hs smaon for h nropy producon s analogously o (7), = R + F Accordng o (6) s ncssary o sma R n h followng qualy: Vh ª-( h - Vh + log m) R

12 Hr h ª ( h + log m ) F ª l Vw V, - ha corrsponds o h nropy flucuaon a ranson Vw = nf w-w, w Œ w, w > w { } In h frs nfnsmal ordr w oban: F R, w -mnmal spcral gap log m -h lw (log m -h) ª ª h + log m Vw w w, w > w (8) For valdy of lnar approxmaon for a local nropy s voluon w assum wha: h + logm > I s nrsng o compar hs valu o un F R F R :, log m + log m ª h >, logm + logm > h (9) Cranly, s a rough sma W s ha rao (9) s drmnd by locaon of h rajcory n h powr spcrum, and by h powr spcrum dnsy h dynamc mod of a fas dffuson xss n our sysm I praccally concds wh h oprang procss () Fguravly spakng, hs mod allows h sysm o absrac from a sudd par of h graph, and concnras s own "anon" and hgh ranson - proprs: D ( dmm) R : o a sarch of low-nrgy aracors From pr-rgodc rasons s possbl o rplac, subgraph: ' " V Œ V : v( V') > v by on vrx wh curvaur (nrgy): ' ' - v = V  v > v, V Œ ' ha s for cohrn vn n sns dynamc facorzaon hs suaon w shall nam h scond ordr dynamc facorzaon hs xampl llusras a prsnc of wo dynamc mods n h sysm h frs, flucuaon s a localzaon of rajcory n h confguraon spac (modul xploraon) h scond on s a rlaxaonal non local (suprmodul xploraon) hs mods rplac ach ohr n such mannr ha h flucuaons drf n a powr spcrum of nvronmn o nrgy mnmum: o flanss W com o h mporan consqunc Proposon-dfnon h Mmory n h sysm: rajcory-nvronmn (E) lads o asymmry of m ha manfss slf as a quas-drcd drf of h masabl vns n a drcon of nrgy rducon-h opologcal convrgnc:

13 v > v > () Proof Frs, a squnc () s nonmonoonous, s rahr ndncy mag L s consdr h masabl vn v > v A nx masabl vn can hav boh grar and smallr nrgy W shall sma h m + of ranson of h rajcory o h masabl vn: vj < v and h m - of ranson o h masabl vn: vk > v hn w shall compar hm L -momn of h bfurcaon, mn - - =, D +v = nf ( v ) k - v -op spcral gap, : and l: k v > v D +v = v k hn n lnar approach, usng smaon for rlaxaon m (9), w oban: D v =»- > w l ( h+ log m) - R + hs s h lowr bound smaon for h rurn m - of sysm n h powr spcrum I s asy o s, ha for + such lmaon dos no xs: As alrady nx raon can ransla h sysm no h masabl phas For hs purpos s ncssary n ordr, h nghbourhood of h acual aracor (vacuum) conand som aracors (vacua), whch hav smallr nrgs (3) Mor prcsly, l { w v } V - " v Œ V : () v < -vacua s wh nrgs smallr n comparson wh h nrgy of acual vacuum hn a probably ha h rajcory wll rach masabl vn wh smallr nrgy durng h rlaxaon m s: (, ) Ê V V p w R Æ wj wj < w ˆ  Á - > Ë V V : R In hs smaon h Brnoull ss nsad of h Markovan procss ar usd, ha s qu admssbl a h mos gnral assumpons of nvronmn A h sam m, as was bn shown, h probably of occurrnc of masabl vn wh grar nrgy durng hs m ( R ) s srong qual o zro for a non-zro spcral gap Fguravly, n ordr an nrs o h hsory has arsn, s ncssary o parally forg Hr hr s a phnomnon of h opologcal convrgnc; h convrgnc on an aracors spac (), and n h nvronmn-landscap spcra rm opologcal convrgnc w hav n sns of h dynamcal facorzaon of h facor-sysm hs s a dynamcal non-monoonous rducon of paron o rval on h fac, ha h mmory, bng ralzd as a gomrcal (mrc) facor, lads o h opologcal convrgnc, s rmarkabl I s possbl o ll, ha n our sysm h mmory s a sourc of conscousnss of h purpos 3

14 h opologcal convrgnc s no convnonal Acually h sysm s nonsaonary w s no rducd o and h mag of s voluon n h powr spcrum of ponal { } a smpl convrgnc (rgular, wak c) Mor lkly, rmnds a urbuln flow For llusraon, vn n cas of h dgnrad global mnmum ({ } w mn, >, < < nn, > ) ransons bwn s non-conncv componns ar possbl For hs purpos, as s asy o s, s ncssary ha bfor a frs passag m of a global mnmum a condon: l logv > w - (3) was sasfd mn Connung a paralll wh quanum mchancs, s possbl o ll, ha h sysm convrgs n classcal sns, as a squnc of dcohrncs Evn afr a global mnmum s achvd h nrsng dynamcs do no dsappar h Global mnmum wll collaps and a larg rlaxaon may ras h rajcory n h landscap spcrum And scnaro wll rpa: h larg-scal oscllaons 7 Cycls on h Landscap spcrum Inrfrnc Hr w shall quanavly sma h nformaon sns n a phnomnon of h opologcal convrgnc o hs nd w shall consdr h cycls of oprang procss and hr acon n h ral "fas m L: { V} V V Q Q+ = Æ Æ Æ rajcory of oprang procss on h suprmodul spac, h corrspondng squnc of vacua lfms: { } Q Q+ = Æ Æ Æ Unlk a parnal procss, -s a srongly corrlad hs s llusraon of h nformaon mmory Schm of our xampl s followng: w shall consdr a cycl of h oprang procss V (): { } CV (, ) V ÆV Æ ÆV Æ V c N L c - a lngh of a cycl (n fas m)- c N = Â, k k k= -aracors lfms Any cycl s prsnd by h opraor: * C ( V,, ): ( ) Æ ( + ) (cocycl) c c h scnaro can looks so: frs, h rajcory, a h momn s graspd by aracorv, hr lvs n a som m (lfm of masabl vn ( ) ), hn h rajcory lavs h aracor and afr m c coms back agan Durng hs m h sysm can or only rlaxs, or can rs h nrmn squnc of flucuaons and rlaxaons W shall consdr rlaxaon mod, as a mos probabl and smpl Clarly, ha nw vn s lfm ( + ( ) + ) wll dffr from ( ) by vru of m hrogny c 4

15 Morovr, $ : " <, ( + + ) < ( ) c c W shall sma h lngh of "mnmal cycl", n hs sns Fguravly bng xprssd, h rajcory s no say oo long hr whr was rcnly hs crcumsanc pays anon o h anomalous dffuson naur of consdrd sysm E W sar ralzaon of hs plan L's us a lnar approach for lfm smaon: D l» l l ( h log )» l + m Hr, as n (7) c - log m º V å log m For smplcy of smaon, w mplcly assum ha h pc of h rajcory corrspondng o a mnmal cycl wll b unformly dsrbud on all spac of confguraons Obvously, s no absoluly so Howvr, f w consdr a cycl as a cohrn (rlaxaon) phas, ha s h mos probabl, our assumpon s qu jusfd Obvously, l W oban for nw, rlav aracor s lfm: =» Dl l ( oh+ log om ) c c c h» h+ h c= h- ( h+ log m ) c log c o m» log m ( + ) - o, W mus sma c n qualy: = Mor accuraly: c = ( + + ) c Nvrhlss, w nrs h frs ordr of smallnss on hrfor w shall consdr h frs varan Oban: -( h+ log m ) h+ ( h+ log m) + log m - log( + ) or, n frs ordr of smallnss: + log m h log m c c h»»- + c c 5, c» Hr s possbl o no, ha for h bfurcaon momn:, h= log V -( lw ) - (3) As on would xpc, h oband smaon dos no dffr from (9) W go o a concluson ha h m of corrlaon and h lfm ar valus of sam ordr hr dsncon s dfnd by non-unformy of dsrbuon of rajcory s pons n confguraon spac Quanavly hs hrogny can b xprssd n rms: log m,log m

16 W hav smaon for h prod of h mnmal cycl L s nam hs valu h corrlaon m As s asy o s, on a s of h vrxs-aracors vsd n h pas, h oprang procss can b consdrd as a squnc of h cycls closs Such "nrfrnc of flucuaons" can lads o h dynamc facorzaon vn of vns wh a las nrgy a prsn Dffrnly: frqun, fas rajcory rurnngs ar dsroyng h masabl vns ranslang hm no cohrn rank 8 Mmory and Dark nrgy As w can s (3), afr a drmnd lvl of dcohrnc, h blow-up lk xpanson s appar Hr w shall consdr a spculav physcal nrpraon of hs phnomnon n E h xpanson on confguraon spac s: D j l w» D= Hj() D, j H j = lw - Hr - whol m, - m afr prformanc of dualy condon (3), s nror of navy rcangl on Fgur, w - characrsc scal of a symmry brakng, l = l ( h + log m) - sl produc scal (7) For smplcy, w assum ha hs valu do no sgnfcan chang durng lfm (Fgur ) Facor h + log m has an nformaon sns, whch s a dffrnc bwn global and local nropy hs nformaon dfc conrols h acclrad xpanson hus, n our oy modl, h dark nrgy has nformaon naur Dffrnly, w may nrpr h voluon of a sngl unvrs as a squnc of dcohrncs (progrssv nanglmn) (rf) ha producs an ffcv nvronmn w ( ) rnormalzaon No ha - l s a dynamc quany, whch can hav arbrary small valu Is fn unng happns by mans of h prvous hsory 9 Dscusson L us summarz wha w hav achvd so far W hav consdrd dynamcs of sysm E on varous scals, and hav oband followng qualav pcur: h flucuaons of h rajcory s dnsy dsrbuon ar alrnad wh rlaxaons Durng flucuaon, h rajcory s localzd n som vacuum and maks oscllaons around of gradn flow qulbrum, absorbng h nformaon on local gomry of an nvronmn Enropy, hus, vnually, dcrass up o som hrshold dfnd by h bfurcaon condon (3) Furhr h rajcory has blow-up lk unnlng no nghborng aracor-vacuum Afr a flucuaon or a squnc of flucuaons ( solon ) h sysm s n a rlaxaon phas h rlaxaon m dpnds on hababl volum, nropy and sz of acual spcral gap (spcrum dnsy) Frs of all, w ar nrsd n a flucuaon mod Afr bfurcaon, h mrcs conans nformaon on an nvronmn, namly, abou a local curvaur of landscap ponal-h symmry brakng hrarchy 6

17 W hav shown ha h mag of dynamcs n a landscap spcrum s rgular n som nformaonal (nformaonal mmory) sns hs larg-scal opologcal convrgnc s mporan, non-obvous propry, ha gnrad by hsorcally spcfd dynamcs h masur, gnrad by rajcory, s non-sngular and praccally ndpndn from nal condons bcaus of mmory h analog of dark nrgy (Hubbl consan) n a confguraon doman s non-consan and proporonal o h dffrnc bwn global and local landscap nformaon In rnormalzaon doman hs s an analog of h cnral charg ha a masur of ffcv dgrs of frdom In E h rnormalzaon s a procss wh jumps bwn local hors hs sms lk h rnormgroup quanzaon Exploraory naur of E s dologcally rlad o h Cnsus Burau concp [3] h Mmory n E s a subjc (xplorr, no always havng h Anhropc gus) of h mulvrs-confguraon spac Rnormalzd nrpraon of E s jusfyng a vry dns dscruum of h vacua nrgs [] I wll dnsr n h fuur bcaus of shor rncarnaons cycl Acknowldgmns I am graful o Alx Morozov for a hlp on h publcaon of hs papr 7

18 l logv w - l l () + l l() l() l Fgur : Lnar approxmaon l logv () + l + l l l() w - l() l logv l Fgur : Quadrac approxmaon w - l() l() Fgur 3: "Brh" of nw vn: V V +, (smoohd) 8

19 Rfrncs [] L Sussknd, h Anhropc landscap of srng hory, arxv:/hp-h/39 [] A Grasmov, D Lbdv, A Morozov On Possbl Implcaons Of Ingrabl Sysms For Srng hory (Moscow, IEP) IEP-9-4, Dc 989 Publshd n InJModPhysA6: , 99, SovJNuclPhys5:9-95, 99, YadFz5: , 99 [3] Maxm Budav, Mmory n h sysm: rajcory-nvronmn arxv:78483v [cond-masa-mch] [4] Bn Frvogl, Lonard Sussknd, A Framwork for h Landscap arxv:/hp-h/4833 [5] GulnD al Dcohrnc and h Apparanc of a Classcal World n Quanum hory (Brln: Sprngr, 996) [6] Gll-Mann M, Harl J B "Quanum mchancs n h lgh of quanum cosmology", n Complxy, Enropy, and h Physcs of Informaon (Sana F Insu Suds n h Sc of Complxy, Vol 8, Ed W H Zurk) (Rdwood Cy, Calf: Addson-Wsly Publ Co, 99) [7] Andr Lnd, Alxandr Wsphal Accdnal Inflaon n Srng hory arxv:/abs/76 [8] Andr Lnd owards a gaug nvaran volum-wghd probably masur for rnal nflaon arxv:/abs/756 [9] Andr Lnd Snks n h Landscap, Bolzmann Brans, and h Cosmologcal Consan Problm arxv:hp-h/643v3 [] A Lnd Inflaonary Cosmology, arxv:/hp-h /7564 [] S W Hawkng, Volum Wghng n h No Boundary Proposal, arxv:79v [hp-h] J B Harl, SW Hawkng & Hrog, h No-Boundary Masur of h Unvrs, arxv:7463v [hp-h] [] Bn Frvogl, Mahw Klban, Marıa Rodrıguz Marınz and Lonard Sussknd Obsrvaonal Consquncs of a Landscap, arxv:/hp-h/553 [3] L Sussknd, h Cnsus akr's Ha arxv:79v [hp-h] 9