On the Hubbard-Stratonovich Transformation for Interacting Bosons

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Pierce Harvey
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1 O h ubbrd-sroovh Trsformo for Irg osos Mr R Zrbur ff Fbrury 8 8 ubbrd-sroovh for frmos: rmdr osos r dffr! Rdom mrs: hyrbol S rsformo md rgorous osus for rg bosos /8

2 Wyl grou symmry L : G GL V b rrso of om L grou G o vor s V Th hrr χ h W G T / Z G χ g : Tr χ g h g V T : χ g s rdl fuo : g h G Thrfor h rsro o mxml orus T G s vr udr h o of h Wyl grou χ w w g w g w /8

3 Wyl hrr formul 3/8 s s SU Sym / l l 3 θ θ θ θ θ θ θ θ χ χ > S S S S m m S W S V G : Exml : y rrdubl rrso o hs For hghs wgh Wyl grou osv roos

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5 ubbrd-sroovh Mhod for Frmos 5/8

6 s Sg 6/8 γ δ β βγδ δ γ β β β β V T V T f f f f v V T V T S f o fors grl u h fuol To s body ro : wo mlo k rgy lus fors : h voluo oror of Mrx lms

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8 Frmos: vr of h mhod 8/8 Tr / / γ γ β β whr Sroovh doulg s bfor : ubbrd s obd by rlg Th xo Grssm flds : ovr grl Wr h ro fuo s fuol m m m m m m W W d

9 modl for bosos rooy 9/8 < Im R X X X X X X X P P Q P Q ν ν ν ν oo : ss : ohr boso Us h mhod of sbly for mlo boso r hrg

10 Whr s h dffuly? L < h ro b of mxd y : ν > rv Th h udr form rulsv X ν X X X s of df sgur X ubbrd Sroovh?: Y X Y d 3 Y ll of h dsrd rors for h gro dom o b m 3 > /8

11 yrbol ubbrd-sroovh Trsformo Md Rgorous Y Fyodorov Y W MRZ rxv:8496 Fyodorov J Phys odsd Mr 7 5 S95 W & Fyodorov J Phys /8

12 /8 kgroud & movo osdr rl symmr mrs Tr / Tr GOE Tr K b K K d μ Us o ob Exo of rrol of rodu of hrrs olyomls: < > Im Im for d for wh \ GOE / F sg Im x / / s d s s π / Tr / b s d s F π

13 kgroud & movo oud s o : ss wh s dg Id Id s vr udr ougo by g sg s O " hyrbol symmry" Wgr 979 x s s ubbrd-sroovh doulg: Tr Tr R R Tr dr Problm: how s hs do orrly?! ss fors R srs bu Tr R Tr RsR s s of df sg om of Shäfr Wgr 98 s o O vr 3/8

14 4/8 Prusk-Shäfr dom < > sv v sv v o s-lk m-lk : v v R O s gvlu lk m d lk s hs mrx dgolbl Evry R 3 losur by dom h mof wh sso g mof ordrg by Eod R o o o O dmso o hs for vr s Eh doms of uo s dom Shäfr Prusk U

15 Sm of rsul L dr dr Lbsgu msur Thorm FWZ Thr xss som ho of uoff fuo R χ R ovrgg ows o uy ho of suh h sg fuo sg lm sg holds ru for ll mrs s s { ± } Tr R s d uu d umbr Tr R χ R dr wh h rory s > Tr Rmrk sg s h ry of h umbr of rsosos o dd o rdu o h xrml form o o o Th roof s o by omlg h sur d shfg 5/8

16 6/8 Id of roof vshs Show h osdr τ ι χ χ R R dr R I d d dr R I sg lm sg lm : Tr Tr & & Rmrk 7 Fyodorov W 5 Fyodorov : :

17 Two doms for R r r r r Lgh-o oords: ξ r η r r r / r / r 7/8

18 Rorgo of boudry omos Rorgo of h boudry s gvs wo ossly ord boudry ls E d F Th grls log E d F r xol d osllory d vsh h lm 8/8

19 9/6 orollry lm sg sg FWZ Tr Tr SO Tr g g d J dg g g s s J J J J > χ hvwodo h ssfs f Th by sg - lrg Job f orollry SO SO dg < J dg d d dg d J dr g g g R Job : for msur r d whr lm Volum d wh L gvlus : of rms Formulo

Special Curves of 4D Galilean Space

Special Curves of 4D Galilean Space Irol Jourl of Mhml Egrg d S ISSN : 77-698 Volum Issu Mrh hp://www.jms.om/ hps://ss.googl.om/s/jmsjourl/ Spl Curvs of D ll Sp Mhm Bkş Mhmu Ergü Alpr Osm Öğrmş Fır Uvrsy Fuly of S Dprm of Mhms 9 Elzığ Türky