Jagiellonian University in Krakow

Transcription

1 Jagllonan Unvsty n Kakow Faculty of Physcs, Astonoy and Appld Coput Scnc Pot Sant Path ntgal appoach to a spl quantu chancal syst bachlo thss wttn und supvson of Pof Jack Wosk Kakow, ()

2 Contnts Intoducton h path ntgal Fynan s knl h Wck otaton 3 Avags of physcal quantts 4 h pont-splttng pscpton Nucal valuaton of path ntgals 3 Calculatng path ntgals usng Mont Calo thods 3 h Mtopols algoth 33 Estaton of statstcal os Rsults fo haonc oscllato 4 Unts and scals of th pobl 4 halzaton, autocolaton and ctcal slowng down 43 Fundantal obsvabls x, p 44 h gound stat of th syst 45 wo-pont colaton functons 45 Dfnton and laton to th hgh ngy stats 45 h fst xctd stat of th haonc oscllato 453 h scond xctd stat Suay 3 Acknowldgnts 4 Rfncs ()

3 Intoducton h path ntgal foulaton of quantu chancs was dvlopd n 948 by Rchad Fynan and playd a cucal ol n pogss of thotcal physcs h path ntgal appoach to quantu chancs, psntd n dtals n book [] by RFynan and AHbbs, vals conncton btwn th lgant Lagang appoach to classcal chancs and th quantu chancs Futho, t povds ntutv nsghts nto bhavo of quantu chancal syst fo t shows that th atx lnt of voluton opato can b calculatd as su ov all tajctos connctng th statng and fnal stat of th syst Nucal coputaton of path ntgals n agnay t foals wth us of Mont Calo tchnqus povd to b an ffcnt way of solvng non-ptubatv quantu pobls, ld to dvlopnt of th lattc fld thoy and bco a powful tool n nvstgaton of QCD In ths thss w ploy th nucal thod to valuat path ntgals n a spl contxt of on dnsonal quantu chancs, whch alady xhbts popts that a ncountd n study of o advancd thos M Cutz and B Fdan consdd anhaonc oscllato n [3] n a sla fashon In [6] a study of th dffnt Mont Calo algoths s psntd Plan of th thss s th followng W stat wth a bf ntoducton of th path ntgal foals, pfo tanston to Eucldan t n scton Moov, latonshps btwn opatos and functonals, ssntal n ou consdatons a xand and dvd In scton 3 w psnt th nucal thod of valuatng th path ntgals hn, n scton 4, th psntd tchnqu s appld to th cas of haonc oscllato (3)

4 h path ntgal Fynan s knl A wav functon of a sngl on dnsonal non-latvstc quantu patcl subjctd to th potntal V (x) s a soluton of th Scho dng quaton: wh th Haltonan H = tb > ta t s gvn by: p ψ(x, t) = H ψ(x, t) t () + V (x ) If at t t = ta th wav functon s ψ(xa, ta ), thn at t ψ(xb, tb ) = dxk(xb, tb ; x, ta )ψ(x, ta ), () wh th knl functon K(xb, tb ; xa, ta ) can b calculatd as th Fynan s nownd path ntgal: K(x, tb ; xa, ta ) = Dx(t) xp Dx(t) xp S[x(t)] = x (t) V (x(t)), (3) xb xa xb xa th acton S[x(t)] s coputd fo a tajctoy x(t) connctng ponts (xa, ta ) and (xb, tb ) h path ntgal (3) s constuctd as follows: th t ntval [ta, tb ] s splt nto N stps of lngth and th bgnnng of ach t stp s dnotd as t ( =,,, N ), a poston x s assocatd wth ach t t and ponts (x, t ), (x+, t+ ) a connctd wth staght lns Povdd that x = xa and xn = xb, a dsctzd tajctoy xn (t), whch conncts ponts (xa, ta ) and (xb, tb ), s dfnd n ths way th valu of acton s coputd fo th tajctoy obtand fo th pocdu of t dsctzaton:! tb N x x j+ j x (t) V (x(t)) = V (xj ) + O( ) (4) S[xN (t)] = dt ta j= and on clas that th popagato fo th tajctoy s gvn by: N KN (xb, tb ; xa, ta ) = π xb xa NY j= dxj xp S[xN (t)] xb xa DxN (t) xp S[xN (t)] (5) n th lt N, whch w f to as th contnuu lt, th dsct t tajctos can appoxat any physcal tajctoy x(t) connctng ponts (xa, ta ) and (xb, tb ), thus th path ntgal s dfnd as th followng lt: Dx(t) xp S[x(t)] = l DxN (t) xp S[xN (t)] (6) N xb xa xb xa h ntgaton n (5) s cad out ov all ntdat ponts x,, xn, consquntly th path ntgal can b ntptd as a su ov all paths connctng th ponts (xa, ta ) and (xb, tb ) (4)

5 h Wck otaton h Wck otaton s a pocdu of tanston to agnay t and povs to b ssntal n nucal calculatons of path ntgals h Eucldan t s ntoducd as: τ = t, (7) usng th abov quaton n th dfnton of acton on fnds:!! tb τb dx dx dt dτ S[x(t)] = V (x(t)) = + V (x(τ )) SE [x(t)], dt dτ ta τa (8) th last qualty dfns th Eucldan acton SE Ral valus of τ wll b consdd, whch ans that w wll b dalng wth th voluton of a quantu syst fo agnay valus of t t h xponnt S[x(t)] - a coplx-valud and oscllatoy functon of S[x(t)] bcos SE [x(t)] - a al-valud and xponntally dapd functon, whch s a dsd fatu n nucal coputatons Anoth plcaton of th tanston to agnay t s th fact, that th voluton of a quantu syst s no long untay Dnotng by n th n-th gnstat of th Haltonan H, H n = En n (En+ > En ) and assung that th syst has dsct ngy spctu, w fnd: ψ(τb ) = H (τb τa ) ψ(τa ) = H (τb τa ) = cn n = En (τb τa ) cn n, (9) = wh En s th ngy of th stat n Unlss th stat ψ s othogonal to, t s pojctd to vacuu fo τb + In od to splfy notaton w wll dnot th Eucldan t by t and th Eucldan acton by S n th st of ths pap 3 Avags of physcal quantts Ou pupos s to valuat path ntgals nucally, w thfo stct ouslvs to th dsct t tajctos xn (t), snc lt N cannot b pfod on a coput W ntnd to xtact physcally anngful nfoaton about a quantu syst, thus a nd to consd opatos psntng physcal quantts n quantu chancs, whch a latd to functonals valuatd fo dsctzd paths xn (t), ass Lt us dfn th patton functon as: () N dx DxN (t) xp S[xN (t)] DxN (t) xp S[xN (t)], x x th ntgaton s cad ov paths connctng ponts (x, t = ), (x, t = ) and th sult s ntgatd ov x, th last qualty n () splfs th notaton h patton functon () s latd to th voluton opato U (, ) = H known fo th Scho dng foulaton of quantu chancs n an ntstng way, whch wll b now shown h fst stp s to not that th acton S[xN (t)] can b wttn as a su of ts dpndng only on th valus of vaabls on th nghbong t slcs:! NY N N = DxN (t) xp S(x+, x ) = DxN (t) xp S(x+, x ), () = = (5)

6 wh S(x+, x ) = (x+ x ) + V (x+ ) + V (x ) + O( ) () W now assu that th xsts an opato on th Hlbt spac of stats : H H such that th atx lnt of btwn gnstats of th poston opato x ( x x = x x) ads: hx x = xp S(x, x ) (3) π h atx (3) s known as th tansf atx Eployng th ontu opato p as a gnato of tanslatons, quaton (3) can b wttn as: hx x = π d V (x) hx p x V (x ) (4) Howv, th last quaton ans that can b xpssd as a functon of th ontu and poston opatos: V (x ) d p V (x ) (5) = π Calculatng th gaussan ntgal ov and tansfong th sult wth th ad of th Bak-Hausdoff foula, on fnally obtans: V = (x ) p V (x ) = p +V (x ) +O( ) = H +O( ) (6) Equatons () and (3) ply that N = N, (7) oov, usng (6), w fnd th dsd latonshp btwn th voluton opato and th patton functon: (8) l N = H N W now pocd to dfn th avag valus of physcal quantts Lt A[xN (t)] b a functonal valuatd fo dsctzd tajctos In fact, th functonal A[xN (t)] s a functon of valus of postons of th patcl x at ts t : A[xN (t)] = A(x,, xn ), (9) snc x,, xn spcfy th dsctzd tajctoy h avag valu of th functonal A[xN (t)] s dfnd as:! N (xj+ xj ) ha = DxN (t) xp + V (xj ) A[xN (t)] () N j= At ths pont a quston about a physcal quantty psntd by th functonal A[xN (t)] gs h pocdu, that was cad out n od to fnd th latonshp btwn th patton functon and th voluton opato, can b convnntly gnalzd n od to fnd an opato foula fo ha hn, opatos, whch appa n th opato foula, wll val th physcal sgnfcanc of A[xN (t)] 8537(6)

7 Lt us consd an xapl of a local functonal A[xN (t)] = f (xk ) wh f s a sooth functon (w wll b patculaly ntstd n th cas of f bng a polynoal) h path ntgal () can b dcoposd as: N π DxN k (t)dxk Dxk (t) NY k Y S(x+,x ) S(xk+,xk ) f (xk ) S(x+,x ), () = =k+ howv, w alady know how to xpss th ntgaton ov x, x, xk and xk+,, xn n ts of opatos Hnc, on nds to pfo th ntgaton ov xk Assung that th xsts an opato O : H H whch fulflls th qunt: xp S(x, x ) f (x ) () hx O x = π and patng th stps takn n quatons (4), (5) w fnd that: O = p +V (x ) +O( ) f (x ) (3) Consquntly, th sultng opato foula ads: hf (xk ) = N k f (x ) k, N (4) whch n th contnuu lt ylds hf (x(tk )) = H ( tk ) f (x ) H tk, (5) wh tk = k hfo, th latonshp btwn th opatos and th functonals, whch dpnd only on on vaabl xk, s staghtfowad Futho, t s ntstng to not that th tac n (5) s coputd fo th opato whch s th poduct of f (x ) n t tk (cf Hsnbg pctu of quantu chancs) and H, whch pojcts to vacuu fo h cas of functonals dpndng on two vaabls xk and xk+ povs to b o coplcatd W now consd a functonal p[xn (t)] = xk+ xk, whch ss to b a good canddat fo a quantty latd wth th ontu of a patcl Dcoposng th path ntgal as n q () and assung that th xsts an opato O : H H whos atx lnts a x x xp S(x, x ), (6) hx O x = π w obtan th followng laton O = V (x ) π p V (x ) d (7) Substtutng = + p and pfong th ntgaton π ( p ) p p d = d = p p, π (8) (7)

8 w fnd that th functonal p[xn (t)] ndd cosponds to th ontu at t tk, snc ts xpctaton valu ads: N k k hp = p, (9) N Encouagd by (9) on ay attpt to gnalz th da and consd a functonal p [xn (t)] = xk+ xk (3) whch looks lk a asonabl canddat fo a quantty assocatd wth th squa of th ontu Agan, w ntoduc an opato O 3 actng n th Hlbt spac of stats H, wth atx lnts gvn by: x x hx O 3 x = xp S(x, x ), (3) π whch, n ts of th ontu and poston opatos, can b xpssd as V (x ) d p V (x ) O 3 = π (3) Substtutng = + p and pfong th ntgaton n q (3):! p p d = d p = p p π π (33) w fnd that th sult dvgs as h fnal opato foula ads: * + xk+ xk N k p k = (34) N h squa of th ontu s an potant obsvabl, snc t s popotonal to th kntc ngy t n th Haltonan, howv, th poposd fo of functonal (3) povs to b ncoct snc sults whch a dvgng as a physcally annglss 4 h pont-splttng pscpton h pont-splttng pscpton s a way of dfnng a functonal cospondng to th squa of th ontu, whch ognats fo th pap [5] of J Schwng h followng functonal pp S = xk+ xk xk xk s consdd as th nw canddat fo a quantty psntng th squa of ontu h path ntgal: xk+ xk xk xk hpp S = DxN (t) xp S[xN (t)], N can b coputd f t s dcoposd as (8) (35) (36)

9 hpp S = N π N xk+ xk Y S(x+,x ) DxN k (t)dxk dxk Dxk (t) S(xk+,xk ) =k+ S(xk,xk ) k xk xk Y S(x+,x ) = (37) Indd, w alady know that ntgaton ov Dxk (t) and DxN k (t) ylds k and N k spctvly, whas ntgaton ov dxk and dxk sults n p hfo, th appopat opato foula ads: N k k hpp S = (38) p p N Calculatng th coutato: h h H +O( ) V (x ) + V (x ) H +O( ),, p = H +O( ), p = (39) w fnally av at th foula: N k p + O() k, N (4) H ( tk ) p + O() H (tk ), (4) hpp S = whch, n th contnuu lt, ads: hpp S = h nus sgn n (4) ognats fo th tanston to Eucldan t Concludng, th pont-splttng pscpton (35) nabls us to calculat th xpctaton valu of th squa of ontu (9)

10 3 Nucal valuaton of path ntgals 3 Calculatng path ntgals usng Mont Calo thods W a ntstd n coputng th avag valu of a physcal quantty psntd by a functonal A[xN (t)], thfo q () s th cntal pont of ou ntst h path ntgal n () can b sply vwd as a hgh-dnsonal ntgal ov th vaabls x, x,, xn Howv, th ntgaton can b pfod analytcally only n a fw cass Hnc, a nucal thod whch woks fo an abtay potntal V (x) s ployd A Mont Calo thod s usd n od to calculat th path ntgal () hus th task s to gnat an nsbl of paths xn (t) so that th avag valu ha s qual to th followng lt: ha = N Nswp DxN (t) xp S[xN (t)] A[xN (t)] = lnswp A[xN (t)], Nswp = (3) wh xn (t) s th -th out of Nswp gnatd paths h quaton (3) holds only f pobablty dstbuton n th nsbl of paths s gvn by: xp S[xN (t)] (3) P[xN (t)] = N W wll now addss th quston of gnaton of th nsbl of paths In od to fnd paths that a dstbutd accodng to (3) w fo a Makov chan of tajctos - w stat wth an ntal N (t) In th Makov chan tajctoy xn (t) and gnat subsqunt paths xn (t), x3n (t),, xnswp th tajctoy xn (t) s obtand fo th pvous on x (t) accodng to a ctan algoth, such a N pocss s chaactzd by [xn (t) xn (t)] - th pobablty of gnatng tajctoy xn (t) f statng fo xn (t) Snc [xn (t) xn (t)] s th tanston pobablty, t satsfs th followng qunts: [xn (t) xn (t)] and DxN (t) [xn (t) xn (t)] = (33) If, aft k stps of th Makov pocss, pobablty dstbuton n th nsbl of paths s gvn by Pk [xn (t)], thn t wll b gvn by: Pk+ [xn (t)] = DN x(t) [xn (t) xn (t)]pk [xn (t)], (34) aft anoth stp of th Makov pocss Snc ou goal s to gnat paths that a dstbutd accodng to (3), w pos th followng qunt on th tanston pobablty: [xn (t) xn (t)] P[xN (t)] = [xn (t) xn (t)] P[xN (t)] (35) h condton (35) s known as th dtald balanc condton and t s a suffcnt condton fo P[xN (t)] to b an qulbu dstbuton of th Makov pocss, bcaus: DxN (t) [xn (t) xn (t)]p[xn (t)] = P[xN (t)] DxN (t) [xn (t) xn (t)] = P[xN (t)] (36) Havng statd fo an abtay tajctoy, aft suffcntly lag nub of stps, th qulbu dstbuton of th Makov pocss wll b achd: P P P P 38383() (37)

11 Snc w a ntstd n asung A[xN (t)] on tajctos dstbutd accodng to (3) and th Makov chan was statd fo an abtay tajctoy xn (t), th suaton n (3) cannot nclud a ctan nub of ntally gnatd tajctos hfo, a ctan nub of stps of th Makov pocss s pfod and only whn th qulbu s achd and tajctos a bng gnatd wth pobablty (3) do w stat to asu ou obsvabl h pocss of appoachng th qulbu dstbuton s calld th thalzaton pocss 3 h Mtopols algoth h Mtopols algoth s an algoth that pfos on stp of th Makov pocss consdd n 3, t updats a tajctoy xkn (t) to obtan a nw tajctoy xk+ N (t) k h tajctoy xn (t), dfnd by th squnc of nubs (x,, xn ), s updatd gadually, n on stp of th algoth only on valu of xj s odfd, whl th st ans unchangd h stp of th algoth s accoplshd n th followng way: A ando nub ξ s gnatd on ntval (, ) wth unfo pobablty dnsty Poposd nw valu of xj s st as xj = xj + ξ wh s a fxd paat A tajctoy xk N (t) dfnd by (x,, xj, xj, xj+,, xn ) s xand k Dffnc btwn th valus of actons fo th tajctos xk N (t), xn (t): k δs = S[xk N (t)] S[xN (t)] (38) s calculatd and f δs < thn th tajctoy xk N (t) s accptd If δs thn a ando nub s gnatd wth unfo dstbuton on th ntval (, ) h vaabl xj s changd to xj f δs > Othws, th pvous valu of th vaabl xj s stod Aftwads, th algoth pocds to th nxt lattc st xj+ h pocdu s patd untl all of th lattc sts (x,, xn ) a pobd and th nw tajctoy xk+ N (t) s obtand anston pobablty of a sngl updat of xj of th algoth ads: k k δs (39) j [xn (t) xn (t)] = n, It s asy to s that: k k j [xk N (t) xn (t)]p[xn (t)] = n S[xkN (t)], S[xk N (t)] k = j [xkn (t) xk N (t)]p[xn (t)], (3) k thus th tanston pobablty j [xk N (t) xn (t)] of on stp of th algoth fulflls th qunt (35) and as a sult, th tanston pobablty assocatd wth th updat of th whol tajctoy k [xk+ N (t) xn (t)] fulflls th dtald balanc condton hus, onc th thalzaton pocss s copltd, subsqunt tajctos gnatd by th Mtopols algoth a dstbutd accodng to (3) Snc n a sngl stp of th Mtopols algoth only on vaabl xj s changd, th xpsson fo δs (38) splfs to:! x x x x j+ j j j k δs = S[xk + V (xj ) V (xj ), (3) N (t)] S[xN (t)] = k whch ans that coputaton of global valus of acton: S[xk N (t)] and S[xN (t)] n ach stp of th algoth s unncssay It s suffcnt to calculat δs accodng to (3) hs spl obsvaton consdably povs pfoanc of th Mtopols algoth 3679()

12 33 Estaton of statstcal os Havng gnatd a st of Nswp paths wth th Mont Calo algoth, w a ady to stat asung th A[xN (t)] Assung that paths xn (t),, xn (t) w gnatd bfo th thalzaton pocss N N Nth + (t) w gnatd by th algoth n had bn copltd, whas tajctos xn (t),, xnswp th qulbu and thus a dstbutd accodng to (3), th followng squnc of asunts s obtand: N th + th (A[xN (t)] : =,, xn, xn,, xnswp ) (3) N N In od to fnd th avag valu ha on ay now coput th athtc avag n (3) Howv, n nucal coputatons on dals wth a fnt Nswp, hnc th nub of asunts Nas = Nswp Nth s also fnt, thfo vfcaton of statstcal os plays a cucal ol n analyss of Mont Calo algoths sults Each valu of A[xjN (t)] (j > Nth ) n (3) cosponds to a ando vaabl Aj Snc tajctos xjn (t) w gnatd by th Makov pocss n qulbu, all of ths vaabls hav th sa xpctaton valu and vaanc: Aj = ha and σa j = σa, (33) wh th ovln dnots an avag ov a st of ndpndnt Makov chans W us th unbasd statos fo ths valus fo on Makov chan: A = Nswp Nas A and σ A = =Nth+ Nswp Nas =N A A (34) th + h stato A s a ando vaabl tslf, as ts valu changs fo on Makov chan to anoth h vaanc of A s gvn by: σa = (A ha) = Nswp Nas (A ha) + =Nth + Nas Nswp (A ha)(aj ha) (35) 6=j,j=Nth + An potant pont s that n th cas of tajctos gnatd wth th Mont Calo algoth, th vaabls A, Aj a colatd snc th succssv tajctos xkn (t), xk+ N (t) a obtand fo ach oth hs lads to non-vanshng autocolaton functon, whch, fo a st Makov chans n qulbu, s dfnd as: (36) CA (t) = (Ak ha) (Ak+t ha) Claly, n od to calculat CA (t) n nucal coputatons, on dos not avag ov th st of ndpndnt Makov chans, but ath uss an stato fo on Makov chan: C A (t) = Nswp t Nas t =N (A A )(A+t A ) (37) th + Usng th dfnton of CA (t) (36) on contnus th calculaton of σa and accodng to [4] obtans:! N as CA (t) σa = + σa + O( ) τa,nt σa, (38) CA () Nas Nas Nas t= wh th last quaton dfns ntgatd autocolaton t τa,nt and th t of od O( N ) s as doppd h quaton (38) has two sgnfcant plcatons h standad dvaton of avag valu of A dcass wth th nub of asunts lk Nas Moov, th statstcal o (38) of th 9537()

13 sult ncass as th autoclatons btwn gnatd tajctos bco stong hs can b ntptd n th followng way: f th asunt was ad on Nas tajctos and th ntgatd autocolaton t s statd to b τa,nt, thn th nub of ffctvly ndpndnt asunts s: Nas (39) Nndp = τa,nt Concludng, th sultng foula fo th avag valu of th obsvabl A ads: σ ha = A ± τa,nt Nas A hs quaton shows th way of quotng sults whch wll b adoptd n scton (3) (3)

14 4 Rsults fo haonc oscllato In ths scton w psnt sults obtand wth th Mont Calo thod ntoducd n th scton 3 fo haonc oscllato h thod woks fo an abtay potntal V (x), w dcd to choos th haonc oscllato, snc analytcal soluton of th pobl s known and thus valdty of ou sults can b asly vfd h Haltonan of th haonc oscllato ads: H = p + ω x, (4) wh ω s th angula fquncy 4 Unts and scals of th pobl h dnsonful quantts appa n th Haltonan of haonc oscllato: [] = kg s, [] = kg, [ω] = s (4) h th quantts (4) dtn th scal of th pobl Solvng th physcal pobl wth a coput dands dfnng dnsonlss quantts, thus th th quantts (4) a cobnd n od to st unts of ass, lngth, t, tc and allow us to substtut dnsonful opatos wth dnsonlss ons: ω x, H H (43) p, x p ω ω Aft ths chang any physcal quantty bcos dnsonlss, fo nstanc, th avag valu of A () n th cas of th haonc oscllato potntal ads:! N (xj+ xj ) ha = DN x(t) xp + xj A[xN (t)] (44) N j= Paats whch occu n ou coput calculatons splt natually nto th goups: physcal quantts: th ass of th patcl nsd haonc potntal wll, th angula fquncy ω and th t of voluton of th syst In unts dfnd by (43) th ass and angula fquncy a sply =, ω = and w a lft wth only on physcal paat - th t paat assocatd wth th dsctzaton of th tajctoy - th nub of t stps N, whch s latd to th sz of a t stp : = N (45) paats of th Mtopols algoth, aong th th nub of gnatd tajctos Nswp s th ost potant, snc t dtns th statstcal o of ou sults (3) 4 halzaton, autocolaton and ctcal slowng down As pontd out n sc 3, havng statd th Makov pocss of gnaton of paths fo an abtay ntal tajctoy xn (t), w nd to stan ouslvs fo asunts untl th thalzaton pocss s copltd, snc only thn th tajctos a gnatd accodng to th dstbuton (3) W obsv th pocss of thalzaton of ou algoth by asung valu of ctan xj on subsquntly gnatd tajctos, as functon of nub of gnatd paths - Nswp, as psntd n Fgu Fgu shows xapls of tajctos gnatd dung th pocss of thalzaton and aft th qulbu was achd (4)

15 xj N swp Fgu : h valu of xj as functon of Nswp h ntal tajctoy s xn (t) = {4,, 4} hs Fgu llustats th pocss of thalzaton, w s claly that th algoth s appoachng vcnty of paths wth th valu of xj btwn and In ths cas th thalzaton pocss s fnshd whn Nswp h paats a =, N = ) ) ) v) Fgu : Paths, x as functon of, gnatd by th Mtopols algoth dung th pocss of thalzaton psntd on th Fgu fo Nswp = ), ), 5 ), v) h tajctos ) and ) a gnatd by th Makov pocss bfo achng qulbu - such tajctos occu wth vy sall pobablty (3) n th Makov chan n th qulbu and vtually do not contbut n asunts of obsvabls h tajctos ) and v) a gnatd by th Makov pocss n qulbu and a typcal tajctos of a quantu patcl (5)

16 It s ntstng to not that typcal tajctos of a quantu patcl psntd n th Fgu a hghly gula, ths paths sbl ath a ando walk than a classcal oton of th patcl hs fact s n agnt wth th quaton (34) whch tlls us that no an of th squa of vlocty xsts at any pont of th tajctoy As pontd out n scton 33, w xpct that obsvabls asud on subsquntly gnatd tajctos wll b autocolatd h ntgatd autocolaton t τa,nt fo an obsvabl A dfnd by q (38) as: Nswp CA (t) (46) τa,nt = + C () A t= povds nfoaton on how stongly th subsqunt asunts a colatd h autocolatons a ndd obsvd n data obtand wth th Mtopols algoth W nvstgat th autocolaton of asunts of th poston opato x, Fgu 3 shows Cx (t)/cx () as functon of t C x H t L C x H L D = 5 Τ x, nt = 33 8 C x H t L C x H L D = Τ x, nt = t t D = 8 Τ x, nt = C x H t L C x H L 3 4 t Fgu 3: Plot psnts Cx (t)/cx () as functon of t th Mont Calo t spaaton btwn subsquntly gnatd paths Paats = and N = 6 a fxd, th paat whch dtns th axu chang of xj n on stp of th algoth s st to 5,, 8 In od to nz th statstcal os on has to nz th data autocolaton, whch ans that th paat ust b cafully contolld dung coputatons Ponts a connctd to gud th y Howv, th ajo pobl s th latonshp btwn τa,nt and N, snc ou a s to pfo calculaton fo possbly lag N Fgu 4 shows τx,nt as functon of N Τ x, nt H N L Τ x, nt H N L 5 Τ x, nt H N L = z = 45(9) 4 = z = 3() = 5 z = 49(8) 5 N N N 8 5 N Fgu 4: Plot psnts τx,nt as functon of lattc sz N fo =,, 5 W s that th autocolaton t τx,nt z s ncasng quckly wth gowng N Functon f (N ) = a (N ) + b s fttd n ach cas, obtand xponnts z a dsplayd on th plots W thfo obsv that th ntgatd autocolaton t τx,nt vas as th z-th pow of th lattc sz N h concluson s that w obsv a phnonon known as th ctcal slowng down [4] Accodng to [4] τa,nt s xpctd to bhav as: τa,nt (N )z (47) fo any obsvabl A h potant ssag of q (47) s that th nucal cost of obtanng sults wth latvly sall statstcal os gows as z-th pow of th lattc sz N 64(6)

17 Fundantal obsvabls x, p 43 In ths scton w a ntstd n avag valus of x, p obtand wth th Mont Calo thod h Mont Calo thod allows us to coput avag valus of obsvabls accodng to q () fo fnt N, oov, th valu of N cannot b vy lag, snc th coputatonal ffot gows as z-th pow of N, accodng to q (47) Consquntly, takng th lt N, s not fasbl Howv, th contnuu lt s pat of th xact soluton of a quantu chancal pobl wth us of path ntgals - cf (6) hus, th cucal quston w hav to addss s how th fact, that w a stctd to study th syst fo latvly sall valus of N, nfluncs ou sults h contnuu lt cannot b pfod ona a coput, howv, on can xan how th sults dpnd on th valu of N and xpct, that f N s st to b lag nough, th nflunc of t dsctzaton wll bco nglgbl Fgu 5 shows coputd valus of th obsvabls hx, hpp S as functons of th lattc sz N 4 < x > = - < p ps > -4 N < x > 4 = - < p ps > -4 N < x > 4 = 5 - < p ps > -4 N Fgu 5: Man valus hx, hpps as functons of nub of lattc sts N calculatd fo =,, 5 and fttd functons f (N ) Indd, th valus of hx, hpps a stablzng wth gowng N Fttng polynoals n f (N ) = 3 = P N N :, (48) wh P a paats of th ft, allows to xtapolat th Mont Calo sults to N Valu of P obtand n ach ft appoxats th contnuu lt valu: ln hx hx CL, and ln hpps hpps CL (7) (49)

18 Obtand n ths ann valus of hx CL and hpps CL a dsplayd n abl W not, that sults fo = and = 5 a n agnt wthn statd statstcal o and th absolut valus a qual to W addss th pobl of anng of th physcal t n th nxt scton abl : Obtand valus of hx CL and hpps CL 5 hx CL 548(47) 49995(6) 59(56) hpps CL 534(69) 4993(3) 4994(3) It follows fo Fgu 5 that pps, as bng o coplcatd functonal than x cf 4, s convgng o slowly to ts contnuu lt valu than x Futho, w s that wth gowng w nd to st bgg and bgg valus of N n od to b abl to xtact th contnuu lt valus of obsvabls It s ntstng to confont ths fact wth Fgu 4 whch shows, that th constant a s dcasng as th valu of N gows Concludng, all th paats n th pobl a coupld and on has to put th sa nucal ffot to obtan physcally sgnfcant sults ndpndntly on valu of 44 h gound stat of th syst h t of voluton of th syst, th only physcal paat n th pobl, plays an potant ol fo t dtns whth th hgh ngy stats a psnt n ou coputatons Calculatng th avag valu of a local functonal A[xN (t)] = A(xk ) n th bass of gnstats n of th Haltonan H w fnd: H ( tk ) H (tk ) H En hn A n (4) A = hn A n = P En ha = N N n= n= n= Egnstats a odd, En+ < En, thus: ha h A, fo +, (4) hfo, calculaton of ha accodng to q () fo suffcntly lag, povds th xpctd valu of A n th gound stat of th syst Lt us now dtn th valu of whch s suffcnt to spaat th gound stat of th syst Dnotng th ngy gap btwn n-th xctd stat and th gound stat as: n = En E, w wt (4) n th followng fo: P n hn A n, (4) ha = n n= n= h ts n th sus n q (4) a xponntally suppssd wth gowng n, thus, th lag th ngy gaps n a, th low valu of s suffcnt to nglct th ts n wth n > W thfo s, that th dffnc btwn ngs of th fst xctd stat and th gound stat - dtns th valu of whch ust b st n od to obtan th xpctaton valu of th obsvabl A n th gound stat of th syst Exact valu of s unknown, snc E and vn E a not dtnd yt Hnc, on cannot a po tll whch valu of s suffcnt to solat th gound stat W can, howv, nvstgat how th valu of t nfluncs th avag valu of ctan obsvabl A and xpct that ha s stablzng as s gowng lag h dpndnc of hh on th valu of wll b xand n ths scton h functonal whch cosponds to th Haltonan H (4) at t tk ads: H[xN (t)] = (8) (xk+ xk )(xk xk ) + xk, (43)

19 wh 6 k 6 N and th pont splttng pscpton (4) s usd n th kntc ngy t h nus sgn n th kntc ngy t s an potant dtal, snc t xpsss th fact that w study voluton of th syst n agnay t - th appas n th scond dvatv d dtx(t) (cf scton 4) In fact, th ngatv valu of hpp S has alady bn obsvd n scton (43) and thus, th Haltonan H (43) s th su of two quantts, both of whch hav th postv xpctaton valu h Mont Calo sults copad wth th an valu of hh calculatd accodng to (4) P a En P En, wh th valus of ngs En = + n a obtand fo th hh = En n= n= wll known analytcal soluton of th t ndpndnt Scho dng quaton fo th haonc oscllato h suaton s pfod analytcally:! d ( +n) ( +n) hh = P ( +n) = Coth (44) ( + n) = Log d n= n= n= <H >H L Fgu 6: Man valu of th Haltonan hh (43) plottd as functon of h d sybols dnot sults of ou coputatons (vy pont psnts th contnuu lt valu of hh fo th cospondng t ) h functon Coth s plottd n blu Fgu 6 psnts hh as functon of h potant ssag s that hh s stablzng as s gowng Futho, th Fgu 6 povds us wth nfoaton about th valu of that s ndd to b st n od to nglct th contbuton of ts assocatd wth hgh ngy stats n (43) h suffcnt valu s appoxatly W a now abl to ntpt sults psntd n abl and lat th wth ngy of th gound stat of th syst h an valu of th kntc ngy n th gound stat of th haonc oscllato can b calculatd accodng to th val tho as: dv (x ) = h x, h p = h x dx (45) whch pls that th avag valu of th Haltonan s qual to: h H = hh = hx = hpp S, (46) fo th suffcntly lag h ngy of th gound stat s qual to E =, whch s known fo th analytcal soluton of th haonc oscllato As pontd out n scton 43 hx CL and hpps CL fo = and = 5 a, wthn statd statstcal os, qual to (o to n cas of hpps CL ) h absolut valus of hx CL and hpps CL fo = a slghtly bgg than whch ans, that th contbuton fo th hgh ngy stats s not ntly nglgbl fo = (9)

20 Concludng, th ngy of th gound stat of th haonc oscllato s obtand and qual, as xpctd, to At ths pont th quston about ngs of th xctd stats can b addssd Havng gnatd sults psntd n Fgu 6 and knowng th latonshp btwn hh and on could ty to ft functon such as: N cut f ( ) = PNcut Pn Pn, (47) Pn n= n= to obtand sults, wh Pn would b paats of ft, qual to sought ngs En, and Ncut an abtay paat Unfotunatly, such a thod dos not wok poply vn fo Ncut = - th fttd functon s hghly nonlna and obtanng valus of hh fo sall wth asonabl statstcal os qus a lot of nucal ffot h soluton of pobl of fndng ngs of hgh ngy stats s psntd n scton wo-pont colaton functons 45 Dfnton and laton to th hgh ngy stats wo-pont colaton functon (o sply two-pont functon) s a quantty whch s valuatd fo asunts of obsvabls at two dffnt ts dung voluton of syst and thfo s dpndnt on th t spaaton t btwn th asunts wo-pont functons pov to b a convnnt way n calculaton of ngs of xctd stats of a quantu syst Lt us consd th two-pont colaton functon of two local functonals A [xn (t)] = A (xk ) A (tk ) and A [xn (t)] = A (x ) A (): (48) DxN (t) S[xN (t)] A (tk )A () = H ( t) A H t A ha (tk )A () = N N h opato foula on th RHS of (48) can b obtand by a staghtfowad gnalzaton of dvaton of (5) Evn though th od functonals A () and A (tk ) n th path ntgal (48) can b aangd, th t succson of functonals ( both a valuatd at dffnt t) pls th stuctu of th opato foula h tac n (48), coputd n th bass of gnstats of th Haltonan, povds nfoaton about ngs: En (E En )tk H ( tk ) H tk ha (tk )A () = hn A A n = hn A h A n, N n= N n,= (49) whch bcos ha (tk )A () = (En E )tk h A nhn A fo + (4) n= h ngy gaps n a th only paats n latonshp btwn valu of th two-pont functon and t t, povdd that s suffcntly lag Lt us assu, that w a ntstd n th ngy of th fst xctd stat If opatos A and A, such that th atx lnts h A and h A a nonzo fo = and a nglgbl fo 6=, w found, thn w would b abl to dtn th valu of, snc on would xpct a spl xponntal dpndnc btwn th valu of th two-pont functon and t t Futho, w not that vn f ts wth h A and h A hav to b takn nto account fo 6=, th contbuton of ths ts bcos nglgbl fo suffcntly lag valus of t, snc > ()

21 45 h fst xctd stat of th haonc oscllato In ths scton th ngy of th fst xctd stat s dtnd W consd a two-pont functon of A [xn (t)] = x and A [xn (t)] = xk Such a choc of obsvabls nabls us to fnd th valu of In ou cas, th two-pont functon, accodng to q (49), can b xpssd as: En (E En )tk h x n (4) Gx (tk ) hxk x = N n,= h opato x nabls us to dtn th valu of fo th haonc oscllato, snc t s a lna cobnaton of caton and annhlaton opatos: x = a + a, (4) wh a and a a dfnd n th standad way fo th haonc oscllato Hnc, th atx lnt that appas n (4) s nonzo only f = n ± If consdd valus of a suffcntly lag to solat th gound stat of th syst, th two-pont functon bcos: (43) Gx (tk ) = (En E )tk hn x = (E E )tk h x n= N = 6 - N = t t Fgu 7: Plot psnts Log (hxk x ) as functon of t fo = and dffnt lattc szs N = 6, 8 Fttd staght lns (44) a dnotd n bluw not that quaton (4) s nvaant und substtuton t tk, whch sults n th o syty of plots (up to dnotd statstcal os) Fgu 7 psnts th logath of Gx (tk ) as functon of th t ntval tk As xpctd, w obsv that th valu of two-pont colaton functon s xponntally suppssd as tk gows hfo, functons fn (t): fn (t) = A t, (44) a paats, a fttd to gnatd n th Mont Calo sulaton data and shown n wh A,, a dsplayd n abl th fg 7 Rsultng valus of, qual to obtand valus of W not that valus of th fst ngy gap fo N = 6, 8, a, wthn statd statstcal os, qual to W cla that th slghtly low than valus of fo N =, 4 a atfacts of th dsctzaton of tajctos Snc valus of fo N = 6, 8, a n agnt w conclud, that th contnuu lt valu of s a wghtd athtc an (wth wghts σ ) of sults fo N = 6, 8, :,CL = 99847(59) (45) h valu of,cl s n agnt wth th valu known fo th analytcal soluton, wthn 3 statd standad dvatons ()

22 abl : Engy dffnc btwn fst xctd stat and gound stat E and ts standad dvaton σ fo = N σ h scond xctd stat W now pocd to fnd ngy of th scond xctd stat of th haonc oscllato Accodng to (4), th valu of can b coputd slaly as n scton 45, povdd that w fnd opatos A and A wth vanshng atx lnts btwn th gound stat and th fst xctd stat: h A, h A A two-pont colaton functon of A [xn (t)] = xk and A [xn (t)] = x s consdd h squa of poston opato x can b xpssd as: x = a + (a ) + a a +, (46) wth th ad of (4) h quaton (46) ans that th atx lnt h x n s nonzo only f n = ± o n = hus, n ou cas, th two-pont functon can b xpssd as: Gx (tk ) = hxk x = (En E )tk hn x = h x + (E E )tk h x (47) n= t 5 5 t 5 Fgu 8: Plot on th lft psnts hxk x as functon of t fo = and N = 8 Plot on th ght shows Log hxk x a, wh a s constant functon dnotd as blu ln on th lft plot h valu of a s qual to squa of th atx lnt: a = h x, and as xpctd (cf (46)) obtand valus of a a, wthn statstcal os, qual to 4 Fgu 8 psnts coputd valus of th two-pont popagato functon of obsvabl x As xpctd, th two-pont functon, aft subtacton of t whch dos not vansh as t bcos lag, hxtk x h x dpnds xponntally on t t and thfo can b usd n od to dtn th valu of h valu of s calculatd slaly as n scton 45 - xponntal functons fn (t) = A t (48) obtand n fttng a qual to dsd a fttd and dsplayd on th lft plot of fg 8 Paats valus of Rsults a psntd n abl ()

23 abl 3: h scond ngy gap and ts standad dvaton σ calculatd fo = L σ Agan, w obsv that sults fo N = 6, 8, a n agnt wthn statd statstcal os, whl valus of fo N =, 4 a slghtly low Calculatng th contnuu lt valu of as th wghtd athtc an of sults fo N = 6, 8,, w obtan:,cl = 9984(38) (49) h valu of,cl s n agnt wth th valu known fo th analytcal soluton, wthn statd standad dvaton (3)

24 5 Suay In ths thss a way to tackl nucally quantu chancal pobls s psntd A slf-contand dscpton of th Mont Calo thod of valuaton of path ntgals s ntoducd It s ployd n od to obtan ngs of low lyng stats of on dnsonal haonc oscllato Ou wok s chaactzd by an phass on pactcal and physcal ssus ncountd n nucal calculaton of path ntgals whch consttut coon pobls that hav to b addssd n cas of sla studs of o coplx thos In th fst pat of ths thss w bfly ntoducd th path ntgal appoach to quantu chancs, whch s quvalnt to th opato tchnqu h fundantal cospondnc btwn quantu chancal opatos and functonals valuatd fo tajctos povd to b non-tval and ld us towads dfnton of th pont-splttng pscpton o ou knowldg, th dvaton psntd n sctons 3 and 4 s novl Aft th tanston to th Eucldan t was pfod, th facto whch wghts tajctos S[xN (t)] n th path ntgal bca S[xN (t)] and thus ad th nucal calculaton of path ntgals fasbl h nxt stp was to not that th path ntgal, n th cas of th t dsctzd tajctoy, s a hgh dnsonal ntgal ov th lattc vaabls and as such can b valuatd wth th Mont Calo thod h Mtopols algoth povd to b both fasbl to plnt and an ffcnt way of gnatng tajctos, howv, n cas of systs wth a non-local acton, pfoanc of th Mtopols consdably dcass and on uss algoths whch bas on dffnt das - fo nstanc th Hybd Mont Calo cf [6] Fnally, havng ppad th nucal thod, w appoachd th pobl of haonc oscllato W w ath ntstd n cophnson of th functonng of th Mont Calo thod than n obtanng xcssvly accuat sults h fst stp of analyss was to nvstgat th pocss of thalzaton of th algoth It was found that asunts of th poston obsvabl on gnatd paths stablz aft a ctan nub of algoth stps Onc th thalzaton pocss was copltd, w w abl to asu physcally ntstng obsvabls avodng systatc os As on would xpct, data obtand fo asunts of an obsvabl on subsquntly gnatd tajctos w autocolatd hs autocolatons w gowng stong as th lattc sz N was ncasng, oov, th ntgatd autocolaton t tund out to b popotonal to th pow of th lattc sz: τa,nt (N )z, whch ans that w ncountd th ctcal slowng down of th algoth hn, th quston of pact of th fnt N on ou sults was addssd Rlatonshp btwn th an valus of obsvabls x, p and th lattc sz was xand, th sultng avags hx and hp P S w found to b convgng to th contnuu lt valus, whch a of physcal sgnfcanc h nxt stp was to xplo th dpndnc of t of voluton of th syst on ou sults W hav shown that th contbuton of hgh ngy stats n avags of obsvabls s dcasng as s bcong lag and that th valu of suffcnt to solat th gound stat s dtnd by th fst ngy gap Study of th xpctaton valu of th Haltonan hh as functon of ndcatd, that s nough to cov th avag valus n th gound stat of th syst Evntually, th two-pont colaton functons w dfnd and w found that th colatos of x and xˆ a convnnt tool n dtnaton of ngs of th fst and th scond xctd stat of th syst Concludng, ths thss donstats that path ntgals can b ffctvly valuatd wth us of Mont Calo thod h an advantag of th thod s t staghtfowad xtnson to systs wth o dgs of fdo A fld thoy s obtand whn th t slcng s placd by a spac-t lattc Futho, th thod can b ployd n nvstgaton of non-ptubatv pobls, whch play a cucal ol n QCD Hadon spctoscopy s an xapl of such a pobl Hadon asss can b obtand wth us of th two-pont functon n th sla ann (cf []) as ngs of th fst and th scond xctd stats w obtand n ths thss (4)

25 Acknowldgnts I want to xpss y gattud to y supvso, Pof Jack Wosk, fo any usful dscussons whch dpnd y undstandng of path ntgals and suggstons whch hlpd to pov ths anuscpt Moov, I would lk to thank Andzj Sywd fo th njoyabl t w spnt n uthn lanng th path ntgal appoach to quantu chancs (5)

26 Rfncs [] Fynan RP, Hbbs AR, Quantu Mchancs and Path Intgals, McGaw-Hll, Nw Yok 965 [] Gattng C, Lang CB, Quantu Choodynacs on th Lattc, Spng, Bln Hdlbg [3] Cutz M, Fdan B, A Statstcal Appoach to Quantu Mchancs, Ann Phys 3, (98) [4] Monvay I, Mu nst G, Quantu Flds on a lattc, Cabdg Unvsty Pss, 994 [5] J Schwng, On Gaug Invaanc and Vacuu Polazaton, Phys Rv 8 (95) 664 [6] Sant P, Sywd A, Path ntgal appoach to spl quantu chancal syst, Intnal Rpot, uthn (6)