Geometrical measurements in three-dimensional quantum gravity

Transcription

1 Geometcal measuements n thee-dmensonal quantum gavty John W. Baett axv:g-qc/0008v Feb 00 School of Mathematcal Scences Unvesty of Nottngham Unvesty Pak Nottngham NG7 RD, UK E-mal ohn.baett@nottngham.ac.uk Febuay th, 00 Abstact A set of obsevables s descbed fo the topologcal quantum feld theoy whch descbes quantum gavty n thee space-tme dmensons wth postve sgnatue and postve cosmologcal constant. The smplest examples measue the dstances between ponts, gvng specta and pobabltes whch have a geometcal ntepetaton. The obsevables ae elated to the evaluaton of elatvstc spn netwoks by a Foue tansfom. Dstances In geneal elatvty we can measue the dstance R between a pa of ponts by consdeng the length of a geodesc between them (Fgue ). In quantum gavty the metc fluctuates, so we expect only to be able to say what the possble values fo R ae, and the pobabltes. In geneal we Ths s the fst of thee lectues gven at the Xth Opoto Meetng on Geomety, Topology and Physcs, Septembe 00. Copyght c John W. Baett 00

2 p R q Fgue mght expect these to depend on the topology of the manfold M n whch the ponts p and q le. The Tuaev Vo state sum model [] gves a theoy of quantum gavty n dmensons whee the metc has postve sgnatue []; t gves a concete method fo calculatng the functonal ntegal fo thee-dmensonal gavty []. You can thnk of ths as elated to a -dmensonal theoy wth +++ sgnatue metcs whee the tme dmenson ( ) has been dopped. Although ths s not entely ealstc, t does gve us a model n whch the elatonshp between classcal geomety and quantum gavty can be exploed. The model s specfed by an ntege. Gven ponts p,q M connected by a cuve then an obsevable can be defned whch takes values n the set of spns { 0, },,...,. The pobablty that the spn takes value can be calculated to be P = (dm q), N wth dm q the quantum dmenson of the spn epesentaton of U q sl() fo q = e π/, and N a nomalsaton constant. The fomula fo the quantum dmenson s ( sn π ) dm q = ( ) ( +) sn π andn = (dm q) stheconstantwhchensuesthat P =. Actually n ths model the dstance measuements only depend on the topology of p, q and M to the extent that the ponts p and q ae equed to be n the same connected component of M. The topology of M comes nto genealzatons of the fomula consdeed futhe below. Fst I wll descbe how the pobablty fomula s calculated, and then ts physcal ntepetaton.

3 Calculaton The Tuaev-Vo state sum fo a closed compact manfold M s a fomula fo an nvaant Z(M) R. Ths s defned wth the ad of a tangulaton of the manfold; howeve the value of Z(M) s ndependent of the tangulaton chosen and depends only on the topology of M. k Fgue : State fo a tangle A state fo ths state sum model s the assgnment of a spn,,k,... to each edge of the tangulaton (Fgue ) such that the followng admssblty condtons fo each tangle ae satsfed. +k () k + () k + () + +k () + +k = 0 mod () Gven a state, each smplex s assgned a weght, a eal numbe. Ths numbe depends onthe spn labels fo theedges nthat smplex. The weghts ae calculated usng the spn netwok evaluaton based on the Kauffman backet wth A = e π/ as follows []:

4 Smplex Weght Spn Netwok N dm q dm q = Θ Θ = τ τ = Fo the tetahedal smplex, the spn netwok n the ght-hand column s the gaph whch s dual to the edges of the tetahedon. The state sum fomula s Z(M) = weghts. states smplexes The pobablty fomula s calculated by fndng a tangulaton of M such that p and q ae the two vetces of a sngle edge n the tangulaton. The pobablty P s ust the pobablty that the dstngushed edge has ts spn equal to,.e. P = Z(M,) Z(M), whee Z(M,) s the sum ove the subset of states that have spn on the dstngushed edge. Clealy Z(M) = Z(M,), so the P sum to. The poof that the fomula fo P s coect s to calculate t explctly fo a patcula tangulaton, and then use the tangulaton nvaance of the state sum fomula to show that the fomula holds fo all tangulatons whch have an edge that uns fom p to q. The calculaton can be done easly fo M = S usng the sngula tangulaton of S wth two tetaheda. It also follows fom the Foue tansfom esult poved below. The poof that

5 the esult s the same fo any tangulaton wll appea elsewhee. The fact that the answe s the same fo any manfold follows fom the connected sum fomula fo the Tuaev-Vo nvaant and the fact that the edge s contaned n a ball n M. The postvty of the pobabltes s not mmedately obvous fom the defnton of the state sum snce the total weght fo a state of M can have ethe sgn. Howeve t does follow fom the fact that the Tuaev-Vo model has state spaces on sufaces whch ae Hlbet spaces, and the state sum fomula fo P s the expectaton value of a postve opeato (a poecto) on the Hlbet space of S. Geometcal Models A physcal ntepetaton s most appaent n the lmtng case (the Ponzano Regge model [8]). Then, P = ( +). N Thee s howeve no value fo N whch nomalses P to so ths lmt s somewhat degeneate. Nevetheless, Ponzano and Regge dscoveed that the asymptotc fomulae fo the state sum n the lmt have a geometc ntepetaton f one takes + to be the length of the edge n -dmensonal Eucldean space R. Also, they suggested that the sem-classcal confguatons of the state sum model ae gven by mappng the smplcal complex to R wth an appoxmate unfom measue fo the poston of the vetces n R. Ths s also consstent wth the gauge theoy ntepetaton of the model n whch the gauge goup s the sem-dect poduct of SU() and R []. These consdeatons suggest that fo spn, the dstance between p and q s + and the pobablty P s popotonal to the aea of the -sphee of adus +. In othe wods, a geometc model fo the pobabltes P s to consde dsplacement vectos n R whch have length R = + but undetemned decton. The measue P s a dscete veson of the unfom measue πr dr n thee-dmensonal Eucldean space. Ths gves a model fo the P n tems of pobablty measues fo ponts movng n the classcal geomety. Now to etun to the Tuaev Vo model. The Lagangan quantum feld theoy vew s that ths model s a veson of quantum gavty wth a postve cosmologcal constant Λ, wheeas the Ponzano Regge model has Λ = 0. The classcal solutons ae locally a -sphee, wth adus /Λ. Obvously as Λ ths degeneates to the Eucldean space R of the Ponzano-Regge

6 model. Ths suggests that the physcal ntepetaton of the pobabltes P should be based on confguatons n S. Indeed the aea of a -sphee of adus + n S s aea = πsn π ( +), ().e., popotonal to P, f the -sphee has adus /π. If the pont p s fxed at the noth pole then the possble postons fo q le on the -sphees ndcated on Fgue wth pobablty popotonal to the aea. In the fgue, p =0 =/ = =(-)/ =(-)/ Fgue : Possble obts fo q the -sphee s poected to a dsk on the plane and the -sphee of constant heght s shown n ts poecton as a hozontal lne. In ths way the ange of values fo the spn also has a natual explanaton n tems of the -sphee: the lengths take all possble half-ntege values fo dstances onthe -sphee of adus /π. Ths only woks because of the + n the elaton between spn and dstance. The mnmum dstance s then / and the maxmum ( )/. Thee ae two othe possble half-ntegal values fo the dstance between a pa of ponts, namely 0 and the half-ccumfeence /. Howeve the coespondng pobabltes n ths pctue ae zeo, and so these possbltes don t occu.

7 Genealzatons In a smla way we can calculate the pobablty n the state sum model fo thee ponts p,q,s whch ae the vetces of an embedded tangle n M to be sepaated by dstances +, +, k + (Fgue ). The esult s s +/ p +/ k+/ q Fgue : Geomety fo a tangle P(,,k) = Z(M;,,k) Z(M) { N = dm q dm q dm q k f (,,k) admssble, 0 else. In cayng out ths calculaton, the topologcal confguaton s mpotant. One has to specfy cuves whch connect each pa of ponts. What s mpotant s that the loop of the thee edges s unknotted and s a contactble loop n M, n othe wods that the thee edges do ndeed bound a tangle n M. The non-zeo pat of ths fomula s sn π (+) sn π ( +) sn π (k +) (7) whch s postve, the sgns cancellng due to the admssblty condton (). The othe fou admssblty condtons () () have a geometcal ntepetaton when they ae ewtten n tems of the lengths: + < ( + )+(k + ) (8) + < (k + )+(+ ) (9) k + < (+ )+( + ) (0) (+ )+( + )+(k + ) < () 7

8 The fst thee ae ntepeted as the condtons fo the edge-lengths of a nondegeneate tangle n a metc space geomety. A tangle s degeneate f thee s a vetex whose locaton s unquely detemned by the locaton of the othe two vetces. Howeve the fouth condton s agan specfc to a sphee: a geodesc tangle wth sdes R,R,R on a sphee (n any dmenson) of adus /π satsfes the nequalty R +R + R. The poof of ths s vey smple. The tangle nequaltes fo tqs (Fgue ) gve R (/ R )+(/ R ) o R +R +R. p q R R R s t Fgue : Geodesc tangle on a sphee Howeve nthecaser +R +R = thetheepontsleonadameteand one of the ponts s detemned unquely by the locaton of othe two. Such a tangle s theefoe degeneate. The oveall esult s that the condtons (-) ae the condtons fo a non-degeneate tangle on S. The geometcal model () fo the sngle edge can be extended to ths case. Consde thee ponts p, q and s on S wth a unfom pobablty dstbuton. The pobablty that the dstances between them ae R, R and R, as n Fgue, s popotonal to sn πr sn πr sn πr dr dr dr 8

9 as long as the nequaltes fo a tangle ae satsfed (Appendx ). Ths fomula s the contnuum analogue of (7), and n fact (7) s obtaned by substtutng R = + /, R = + /, R = k + / n ths pobablty densty. Ths means that the geometcal model epoduces the measue P(,,k) unde the addtonal assumpton that all edge length ae equed to be a half-ntege. In a smla way one can analyse an embedded polygon n M, obtanng pobabltes whch can be consdeed as a measue of the volume of confguatons of an unknotted ccula loop of ods of fxed length n S. It s an nteestng poblem to elate ths to othe measues of the volume of these confguatons, such as the symplectc volume measue povded n the flat ( ) case by the Remann Roch theoem [, ]. These smple examples may gve the msleadng mpesson that the classcal geomety s always the standad metc -sphee. Howeve ths s not the case, as the obsevable s senstve to knottng and lnkng. The geneal stuaton s studed n the next secton. Foue tansfom In geneal one can consde the set of edges on whch the spns ae fxed to fom an embedded gaph Γ n M. Then the state sum nvaant wth these spns fxed gves an nvaant of the embedded gaph unde motons of the gaph n the manfold (ambent sotopes). In the case of M = S thee s anothe nvaant of embedded gaphs wth edges labelled by spns, the elatvstc spn netwok nvaant defned byyette [, ]. Inthssectontsshownthatthetwonvaants aeelated by a Foue tansfom of the spn labels. Ths substantally genealses the Z Foue tansfom of [0, ]. The defnton of the elatvstc spn netwok nvaant s as follows. Let Γ(,,..., n ) be a gaph embedded n S, and ts edges labelled wth spns,,..., n (n a fxed ode). Fst, the nvaant s defned n the case of tvalent gaphs, then ths wll be genealsed to abtay vetces. Fo each vetex of a tvalent gaph thee ae thee spn labels (,,k) on the thee edges meetng the vetex. The nvaant s defned to be zeo unless each tple satsfes the admssblty condtons () (). Suppose that these A dffeent set of obsevables to the ones nvestgated hee wee defned n [, ]. 9

10 condtons ae satsfed fo each vetex. Put Θ = Then the elatvstc nvaant Γ(,,..., n ) R s defned n tems of the Kauffman backet nvaant of the dagam gven by poectng the gaph n S to S by k Γ R = Γ vetces Θ. Ths defnton s extended to abtay gaphs by the elatons = dm q R R whch defnes an n-valent vetex ecusvely, fo n >, = δ 0 fo -valent vetces, and k R R = dm q δ k fo -valent vetces. The elaton between the state sum nvaant of a gaph Z(S,Γ) and the elatvstc nvaant Γ R s gven by a Foue tansfom n the spn labels, usng the kenel The esult s Theoem. K b (a) = ( ) bsn π (a+)(b+) sn π (a+). Z(S,Γ(,,..., n )) K Z(S ( )K ( )...K n ( n ) )... n R R = Γ(,,..., n ) R. () 0

11 A geneal poof of ths esult wll appea elsewhee. Howeve I wll pove a patcula specal case whch s nteestng, as the esult mples some new denttes among quantum -symbols (Appendx ). Ths example s also suffcent to povde a poof of the esults fo the edge and the tangle gven eale. The example s the tetahedal gaph embedded n S. The defnton of the state sum nvaant s ( Z S, ) = dm q...dm q N Θ(,, )Θ(,, )Θ(,, )Θ(,, ), () snces canbe tangulated wthtwotetaheda. Thefollowngcalculatons pove the theoem fo ths example. Usng Robets chan mal [9], the squae of the spn netwok evaluaton on the ght-hand sde can be expessed as a lnk dagam n whch some components ae labelled wth the fomal lnea combnaton = (dm q ) of spns.

12 Θ(,, )Θ(,, )Θ(,, )Θ(,, ) = N = N () usng the handleslde dentty fo. The Foue tansfom kenel s elated to the Hopf lnk K () = dm q, and the acton of the Foue tansfom on an edge of a spn netwok s gven by the eplacement K ()dm q =

13 Applyng the Foue tansfom to () gves... Z ( S, ) K ( )K ( )...K ( ) = N 7 = N = N Θ(,, )Θ(,, )Θ(,, )Θ(,, ) = Z(S ) R. () The gaph n the fnal elatvstc spn netwok s the same as the gaph of edges n the ognal patton functon Z. But now the admssblty condtons apply to tples of spns meetng at a vetex of the gaph, wheeas they appled to tples aound a tangula ccut of the ognal gaph n Z. Fom ths example t s possble to pove the theoem vey easly also fo sub-gaphs of the tetahedon. Settng, fo example, = 0 n () gves, on the left-hand sde, a summaton ove weghted wth K 0 (J ) =, whch gves the coect state sum fomula fo the gaph wth ths edge

14 emoved, whlst on the ght-hand sde ths gves the elatvstc nvaant fo the gaph also wth ths edge emoved. The esults at the begnnng of the pape can be checked vey easly. Fo example, the elatvstc spn netwok evaluaton fo s δ 0 and nvetng the tansfom gves P = dm q K ()δ 0 = dm q. Thee s a cuous analogy between the Foue tansfom and the dualty between poston and momentum vaables of a patcle n quantum theoy. In fact the kenel K (a) of the Foue tansfom s a dscete veson of the zonal sphecal functon on S. The Laplace opeato on S (wth adus /π) has egenvalues φ = π ( +)φ fo non-negatve half-ntege ; the egenfuncton that s sphecally symmetc about p S (the zonal sphecal functon) s G (R) = ( ) sn π ( +)R sn πr, whee R s the dstance fom p. Puttng R = a+ shows that at half-ntege values, G concdes wth the Foue tansfom kenel K (a) = G (a+ ) so that the Foue tansfom can be ntepeted as a tanston to a sot of momentum o mass epesentaton fo the quantum pobabltes. Appendx. ponts on S If thee ponts ae dstbuted on S wth unfom pobablty, then ths detemnes a pobablty dstbuton on the space of dstances between these thee ponts. The -sphee has standad sphecal coodnates(χ, θ, φ) whch detemne ponts n S R by π (cosχ,snχcosθ,snχsnθcosφ,snχsnθsnφ). Usng the otatonal symmety, thee ponts on S can be assumed to be at

15 (χ, θ, φ) coodnates The pobablty s thus p = (0,0,0) q = (χ,0,0) s = (χ,θ,0) dp = π sn χ dχ π sn χ snθ dχ dθ. Fo thee ponts on a -sphee of adus /π, the dstances between them (Fgue ) ae gven by R = π χ R = π χ cos π R = cosχ cosχ +snχ snχ cosθ, the last equaton beng the cosne law fo the sphecal tangle pqs wth θ the angle at p. Dffeentatng these elatons gves dp = π sn πr sn πr sn πr dr dr dr when the nequaltes fo a sphecal tangle ae satsfed, and zeo othewse. Appendx. Identty fo -symbols The -symbols ae defned to be nomalsed vesons of the tetahedal spn netwok evaluaton [7]: { } = q Θ(,, )Θ(,, )Θ(,, )Θ(,, ).

16 Usng ths defnton, the dentty poved afte the statement of the theoem s { } { } H( N, )...H(, ) = whee... H(,) = K ()dm q = sn π q (+)( +) sn π ( ) +. The dentty does not appea to have a classcal (q = ) analogue. Refeences [] J. W. Baett, Quantum gavty as topologcal quantum feld theoy. J. Math. Phys. 79 (99) [] J.W. Baett, The classcal evaluaton of elatvstc spn netwoks. Advances n Theoetcal and Mathematcal Physcs 9 00 (998) [] J.-C. Hausmann and A. Knutson, Polygon spaces and Gassmannans, L Ensegnment Mathematque (997), [] M. Kapovch and J. Mllson, The symplectc geomety of polygons n Eucldean space, Jou. Dff. Geom. (99) 79-. [] M. Kaowsk and R. Schade, A combnatoal appoach to topologcal quantum feld theoes and nvaants of gaphs. Commun. Math. Phys. 0 (99) [] L.H. Kauffman and S.L. Lns, Tempeley-Leb ecouplng theoy and nvaants of -manfolds. Pnceton UP (99) [7] A.N. Kllov, N.Yu. Reshetkhn, Repesentatons of the algeba U q (sl()), q-othogonal polynomals and nvaants of lnks. In: Infnte- Dmensonal Le Algebas and Goups. Ed. V.G. Kac, Wold Scentfc. 8 9 (989) [8] G. Ponzano and T. Regge, Semclasscal lmt of Racah coeffcents, n Spectoscopc and Goup Theoetcal Methods n Physcs, ed. F. Bloch, Noth-Holland, New Yok, 98. [9] J. Robets, Sken theoy and Tuaev-Vo nvaants. Topology (99) q

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