V PH V I XNGCLB VLT&TT Phn 1: Gii thiu s Xt trng th c dng nh chic m sombrero, vi i xng quay xung quanh trc thng ng. Mt cht im c in ch c th ti mt im trong khng gian, nn khi n cn bng bn di y trng th, v tr tng i ca n ph v i xng vn c ca h Vt L. Tnh cht th v ny cng c quan st thy trong s chuyn pha ca trng thng k, vi ph v i xng ca trng th tng tc trong nhng khng gian tham s thch hp. Tuy nhin, th gii lng t, ni m gn nh mi tnh cht Vt L u b nhe i bi cc dao ng lng t, mt cht im (bc t do ca l thuyt c hc lng t) v tr cn bng bn s mang nng lng cao hn y ca trng th mt cht (do tnh cht lng t ha). Bng vic chui hm qua li gia cc vng khc nhau pha di cng ca chic m sombrero, nng lng cht im s c th gim xung (vn cao hn gi tr y, nhng thp hn gi tr ca n nu b qua kh nng chui hm), v hiu ng mang tn instanton ny c khm ph nh tnh v nh lng bi Polyakov. S ph v i xng ca trng thi nn l khng cn na, v hm sng (m t xc sut phn b trong khng gian) ca cht im s c li i xng ca h Vt L. Nhng mt ln na, s ph v i xng s xy ra, nu ta s lng cc bc t do trong h Vt L ca ta rt rt ln (tin ti v hn). Ni cch khc, tnh cht qun th ca cc bc t do c th s dn ti s ph v i xng trong th gii lng t, v iu ny thc t xy ra vi l thuyt trng lng t -- chnh l c ch Anderson-Higgs ni ting trong m hnh chun (cho ta ph khi lng cng nh tnh cht ma trn tn x ca tng tc cht ch nhng ti gin m vn ph hp vi cc quan st thc nghim). Trc khi khp li phn bn lun ny, xin cn ch mt iu, l bc t do khng phi l th duy nht thay i tnh cht Vt L khi i t gc nhn c in n gc nhn lng t. Trong l thuyt trng lng t, th mi tnh ton im cc tr u ch l phn ng gp vo l thuyt thng qua nhng gin Feynman bc cy m thi, v khi k n bc vng, th nhiu iu th v c th xy ra. Cc tiu ca trng th c th ban u ch l mt im trn trc i xng, nhng qua nhng b chnh lng t th im ny s tch ra thnh cc im cc tiu khc, gy ra kh nng cho s ph v i xng ca trng thi nn. Hin tng ny chnh l c ch Coleman-Weinberg. M rng hn na, ta thm ch cng khng cn ti th nng s ph v i xng xy ra, v trng thi nn b suy i (nhiu trng thi nn khc nhau cng nng lng). Phn 2: nh hng t s hu hn v v hn ca bc t do Thay v l lun mt cch cht ch v chi tit thng qua l thuyt i s lng t nhn ra vai tr v s lng ca bc t do, chng ta ch xt mt m hnh n gin v ph bin trong Vt L trng thng k cng nh trng lng t -- m hnh kh Bose l tng (i xng

), trong khng gian vi th tch :

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)( )

Halmintonian ca h c th c biu din thng qua s lng cc ht

:

Trng thi nn (tc trng thi mang nng lng nh nht) vi s bc t do(tng s ht) l:

Vi s lng bc t do c nh, th trng thi chn khng ny l duy nht, vic nh. Gi, ta gi nguyn mt ht

v cho th tch ca khng gian chy ti gi tr rt rt ln . Chng ta s cng nhau chng minh rng, trng thi nn lc ny s khng cn ng vi biu din n ca nhm i xng () na. Xt trng thi mch lc:

()

Khi gi tr th tch khng gian tin ti v cng , th cc trng thi

hon ton khng chng nhau na (s dng cng thc Baker-Hausdorf):

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)

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))

iu ny cng dn ti vic tt c cc trng thi trong khng gian Fock

vi chn khng

s vung gc vi mi trng thi trong khng gian Fock

vi chn khng

khi cho gi tr . Mc d trng thi mch lc (theo nh ngha) l vector ring ca ton t hy

, nhng n khng phi l vector ring ca ton t m s ht (mc du vy, c th d dng thy rng s lng ht trung bnh l

), v iu ngha l cc trng thi ny ph v i xng. Tuy vy, bt nh ca mt s lng bc t do(i lng dng kim sot h) li bin mt gii hn th tch ln (s dng k hiuv c tnh ca ta ly gi tr trung bnh ch khng phi chnh xc):

(

)

iu dn n h qu l thuyt (cng quan st c t thc nghim) l trng thi nn ca kh Bose l tng s b tnh cht qun th lm cho suy i, v i xng () s b ph v. Gi, n lc ta cn phi gii thch v sao vi th tch khng gian rt ln th vic chnli mang nhiu ngha Vt L hn l chn . Cu tr li l, do tnh cht phn tch ma trn tn x, hay ni bnh dn hn l nhng s kin xy ra rt xa nhau th khng nn c nh hng mnh n nhau (mt cch gi khc ca tnh cht ny, theo Weinberg, l s co cm ca chn khng). Biu din theo ngn ng ma trn tn x, th khi |

|:

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iu ny l kh hiu nhin nu ta dng chn khng l trng thi

, v phn c m t bi cc khng gian Fock

khc nhau l cc phn khng trng nhau trong khng gian Hilbert. Nhng, d thy rng tnh cht phn tch ma trn tn x ny khng xy ra nu ta s dng chn khng l trng thi

(gi nguyn gi tr tng s lng ht):

()

()

()

()

()

Tnh ton chi tit s ch ra rng, kt qu ny tin n mt gi tr khng i, tc h Vt L s c tnh cht trt t tm xa, v iu ny c ngha l, nu l thuyt ca chng ta phc tp hn, thm cc hng t tng tc vo tc dng ca kh Bose, th mc nng lng ca trng thi

ny s cao hn so vi cc trng thi

. Tt nhin, vic chn gi nguynri chos hp l hn chn thngv nhng l do Vt L v k thut khc na, tuy nhin chng ta khng cn phi dnh vi trang na i su vo chi tit. Cc bn mun tm hiu thm c th tm c v l thuyt thng k trng lng t, cho tnh cht thng ging lng t ca h tng tc c mt qu cao s lm nh ngha s lng ht thay i nh th no (nhng vn ny v cng quan trng khi xem xt nhng th nghim RHIC). Xin kt thc phn ny y bng mt nhn xt, l nhng l lun trn (cht ch m ni) ch ng vi gii hnm thi, nhng nn nh rng ban u chng ta chn h khng tng tc lm mi tnh ton n gin. Nhng trng hp phc tp hn ch ra rng, ch cn s lng bc t do ln hoc th tch khng gian ln, th s ph v i xng c th xy ra (tnh cht qun th ny l lin tc).

Phn 3: nh ngha v ph v i xng trong lng t S vng lng t (biu din thng qua thng tin lng t) ca nhng bc t do rt xa nhau, v tng qut, c th c dng nh ngha s ph v i xng. Tt nhin, hy xt mt VD tht n gin thng qua m hnh Ising (v cng quan trng trong Vt L thng k v Vt L cht rn) vi cc khong cch gia cc mt mng l n v. Nu chn

l trng thi tt c spin cng hng ln v

l trng thi tt c spin cng hng xung, th th trng thi nn c th c biu din l

. Gi, ta xt im 1 v im 2 cch nhau mt khongcng cc im ln cn chng trong khong(gi l vng

v

), th th ma trn mt vi cc trng thi ca chng ln lt l

v

, cn vi vng tng

th ma trn mt l

. Thng qua nguyn l c bn v biu din thng tin, Shanon entropy () cho ta nh ngha thng tin chung gia 2 vng khng gian

v

ny l

. Nu gii hnm

khng tin v mo cho rt c cc trng thi nn, th h c th c coi nh l ang trong trng thi ph v i xng. L lun ny da theo ngoi cho trt t tm xa (ODLRO). Cch hiu ny khng yu cu ta phi tng tng ti bc tranh c in nh chic m sombrero na. ngha Vt L ca n th d nhn hn l ch ni chung chung trng thi nn b suy i (mt pht biu thun ty Ton hc).