Shiva and Kali diagrams for composite quantum particle many-body effects

Transcription

1 1 arxv: v1 [cod-at.es-hall] 26 Feb 2009 Shva ad Kal dagras for coposte quatu partcle ay-body effects M. Cobescot ad O. Betbeder-Matbet Isttut des NaoSceces de Pars, Uversté Perre et Mare Cure, CNRS, Capus Bouccaut, 140 rue de Lourel, Pars Abstract For half a cetury, Feya dagras have provded a elghteg way of represetg ay-body effects betwee eleetary feros ad bosos. They however are qute approprate to vsualze fero exchages takg place betwee a large uber of coposte quatu partcles. We propose to replace the by Shva dagras for cobosos ade of two feros ad by Shva-lke ad Kal dagras for coferos ade of three feros. We also show how these fero exchages forally appear a ay-body theory approprate to coposte quatu partcles. Ths theory reles o a operator algebra based o coutators ad atcoutators, the usual scalar algebra based o Gree fuctos beg vald for eleetary bosos or feros havg strct coutato relatos, oly. PACS uber: y

2 2 Although a large aout of physcal probles deals wth coposte quatu partcles, all textbooks o ay-body physcs [1,2] are restrcted to eleetary feros or bosos, by lack of approprate procedures to properly hadle fero exchages resultg fro the Paul excluso prcple betwee the feroc copoets of these partcles. Ths s why varous sophstcated procedures [3-5] were proposed to ap the orgal fero subspace to a subspace ade of effectve partcles, whch are the take as eleetary, these partcles teractg through scattergs costructed o the eleetary scattergs betwee feros, dressed by a certa aout of fero exchages [6,7]. Fdg these appg procedures soewhat usatsfactory, we decded to face the coposteess of the partcles ts full coplexty. The dffculty s actually double: to properly geerate fero exchages takg place betwee a large uber of coposte partcles, but also to clealy defe what ust be called teracto betwee such badly defed obects whch cotuously exchage ther udstgushable feros. A few years ago, we tackled the splest proble o coposte quatu partcles, aely, partcles ade of ust two eleetary feros [8-10]. We called the cobosos, as a cotracto of coposte bosos. We showed how, through a set of two coutators, we ca reach the 2 2 scattergs descrbg teractos betwee the feros of two cobosos, the absece of fero exchages. Through two other coutators, we ca reach the 2 2 scattergs for fero exchages betwee two cobosos, the absece of fero teracto. By cobg these 2 2 exchages, t s possble to geerate all possble exchages takg place betwee a arbtrary uber of cobosos. These exchages are cely vsualzed through a set of ew dagras [9,10], that we called Shva fro the hdu God, due to ther ultar structure. Lke Feya dagras [1,2], these Shva dagras allow us to readly calculate the physcal effects they represet, through rather sple ad tutve rules. The exteso of ths ay-body theory, orgally costructed for coposte exctos ade of oe electro ad oe hole, to coposte feros ade of three feros that we are gog to call coferos s ot so straghtforward. The fact that, stead of two-ar partcles, we ow have three-ar partcles, leads to a far ore coplex dagra topology for fero exchages. The operator algebra fro whch these exchages follow, s also ore elaborate. Ths pushes us to carefully recosder the reasos for usg

3 3 four coutators the case of coposte partcles ade of two feros, order to possbly exted the procedure to ore coplcated quatu obects. The purpose of the preset letter s to work out fro the very frst le, the structure of the ay-body theory ecessary to descrbe three-fero partcles ad ore geerally -fero partcles. We are gog to show that other dagras, called Kal, are eeded, addto to Shva-lke dagras, to possbly descrbe fero exchages takg place betwee coferos. Also, stead of ust two coutators, we ow eed three coutato relatos, aely, two atcoutators plus oe coutator, to fully cotrol all fero exchages takg place betwee three-fero partcles. By cotrast, the 2 2 scattergs for fero teractos the absece of fero exchages, are stll geerated by two coutato relatos oly. These are two coutators the case of cobosos, ad oe coutator plus oe atcoutator the case of coferos. 1 Shva ad Kal dagras 1.1 Fero exchage betwee two-fero partcles Let us frst cosder coboso ade of two dfferet feros α ad β [9,10]. Two such cobosos ca go fro states ad to states ad, uder a sple exchage of ther feros α, as show Fg.1(a). If a thrd coboso state k s volved, we get the dagra of Fg.1(b); ad so o, as show by the Shva dagra for N-body exchage of Fg.1(c). For a easy exteso to ore coplcated coposte partcles, t s of terest to ote that the star-topology (1b) ca be replaced by the le-topology show Fgs.1(d) or ts syetrcal for show Fg.1(e). These three dagras represet exactly the sae exchage: Coboso has the sae fero α as ad the sae fero β as. I the sae way, the star-topology (1c) ca be replaced by the le-topology (1f) or ts syetrcal for (ot represeted). 1.2 Fero exchage betwee three-fero partcles We ow tur to cofero ade of three dfferet feros (α, β, γ). I a fero exchage, such a cofero ca explode to ether (2+1) or (1+1+1) feros.

4 4 If ust two coferos are volved, we ca oly have the process show Fg.2(a), whch s detcal to the oe of Fg.2(b). Ideed, both coferos explode to (2+1) feros, the coferos ad havg oly oe fero coo. If three coferos are volved, we ca have the three coferos explodg (2+1) feros, as Fg.2(c). We ca also have two coferos explodg to (2+1) feros ad oe cofero explodg to (1+1+1), as Fg.2(d). Fally, we ca have the three coferos explodg to (1+1+1) feros: Ths last case ca be represeted ether a star-topology by the dagra of Fg.3(a), or a le-topology by the dagra of Fg.3(b). We see that the dagras of Fg.2 have a Shva-lke topology, whle the dagras of Fg.3 have a ore coplex ultar topology. We are gog to call the Kal, fro the hdu Goddess, ot as kd as Shva. If we ow cosder exchages betwee four coferos, we ca have processes whch two coferos at least explode to (2+1) feros. These are show by the Shva-lke dagras of Fg.4(a,b,c). We ca also have process whch oly oe cofero explodes to (2+1) feros as the xed Shva-Kal dagra of Fg.4(d). Fally, all coferos ca explode to (1+1+1) feros as the two dfferet Kal dagras of Fg.5. Although soewhat coplcated, these dagras are stll qute ce the sese that the quattes they represet are readly obtaed, as usual, by wrtg the product of the wave fuctos of the coposte partcles o the rght sde ad the coplex cougate of those o the left sde, wth the fero varables read fro the dagra, ad by tegratg over all dub fero varables [9,10]. These Shva ad Kal dagras represet fero exchages takg place betwee coposte quatu partcles. Those are the trcky part of ther ay-body physcs. I addto to these exchages, coposte partcles also teract, a ore covetoal way, through teractos whch exst betwee ther feroc copoets. These appear as addtoal teracto les betwee ay two coposte partcles, as show Fg.6. I the ext secto, we outle how ths cely tutve dagraatc represetato for coposte-partcle ay-body effects, ca be geerated fro hard algebra.

5 5 2 May-body forals for coposte quatu partcles We aga cocetrate o cobosos ade of feros (α, β) ad coferos ade of feros (α, β, γ). These feros, whch ca be electros wth up or dow sp, proto, eutro, valece hole, ad so o, are assued to be dfferet; detcal feros, α β, lke the case of the secoductor trplet tro, wll be cosdered elsewhere. The coutato relatos betwee dfferet eleetary feros read as [a k α, b k β ] ηab = 0 = [a kα, b k β ] ηab, where [A, B] η = AB +ηba, whle η ab = ±1. For electros wth up ad dow sps or secoductor electros ad holes, η ab s equal to 1, whle for electros ad protos, η ab = 1. Fortuately, the key equatos whch cotrol the ay-body physcs of coposte quatu partcles do ot deped o these η ab sce they oly appear through ηab 2, as possble to check. Ths s why we ca, for splcty, cosder that all eleetary fero operators atcoute, eve f they correspod to dfferet quatu partcles. We also assue for splcty, that these coposte quatu partcles are Haltoa egestates, order to for a coplete oralzed bass for 2-fero ad 3-fero states. The creato operators for free ad correlated feros are the sply lked by B = k α,k β a k α b k β k β,k α, (1) a k α b k β = B k α,k β, (2) for cobosos, whle for coferos, ths lk reads F = k α,k β,k γ a k α b k β c k γ k γ,k β,k α, (3) a k α b k β c k γ = F k α,k β,k γ. (4) 2.1 Fero exchages betwee coposte partcles () Coposte quatu partcles ade of a eve uber of feros are kow to behave as bosos, whle those whch are ade of a odd uber, are fero-lke. Ths shows up through the coutato relato of ther creato operators C. It reads [C, C ] η1 = 0, (5)

6 6 wth η 1 = 1 for cobosos lke B ad η 1 = +1 for coferos lke F. If we ow tur to the coutato relato betwee destructo ad creato operators, we ote that [C, C ] η1 actg o vacuu gves δ, v whatever η 1 s. It however appears as atural to take the sae coutato relato for (C, C ) ad for (C, C ). Ths leads us to wrte [8,10] [C, C ] η1 = δ, D, (6) where the operator D, whch dffers fro zero for coposte partcles, s such that D v = 0. A precse calculato shows that the operator D s a su of products lke a a the case of two-fero partcles, whle t also cotas products lke a b ba for three-fero partcles; ad so o for -fero partcles. () To go further, we ote that Eq.(5) leads to [ [C, C ] η1, C ] η 2 = η 1 [ [C, C ] η1 η 2, C ] 1, (7) whatever (η 1, η 2 ) are. Cosequetly, order to have a (, ) syetry, requreet whch ca see as physcally relevat, we are led to take η 2 = 1 for both, cobosos ad coferos. Hoogeety the leads to cosder a secod coutato relato, whch reads as [D, C ] 1 = C D. (8) D reduces to a scalar D (0) the case of two-fero partcles, sce D s (a a, b b) oly, whle t also cotas a operator D (1) (a a, b b, c c) the case of three-fero partcles, due to the presece of operators lke a b ba ther D. Fro Eq.(8) actg o v, t s the possble to show that, for cobosos ad coferos, the scalar part of D s gve by D (0) = (δ, δ, η 1 δ, δ, ) v C C C C v. (9) The frst ter the RHS of the above equato, ust correspods to the scalar product appearg the secod ter, for partcles (, ) ad (, ) take as eleetary. Equato (9) thus shows that D (0) ust correspods to all possble fero exchages takg place betwee two coposte partcles startg (, ) states ad edg (, ) states,.e., dagra lke the oe of Fg.1(a) the case of two-fero partcles ad Fg.2(a) the case of three-fero partcles.

7 7 oly, I the case of two-fero partcles [8,10], coutator (8) the reduces to two ters [D, B ] 1 = B ρ λ ρ ( ), (10) sce coboso ca exchage oe of ts two feros, ρ = α or β, wth coboso, to gve cobosos ad. For three-fero partcles, cofero ca exchage oe of ts three feros, ρ = α, ( ) β or γ, wth cofero ; but t ca also exchage two of ts three feros. Sce λ αβ ( ) s othg but λ (0) γ, as see fro Fgs.2(a,b), the scalar D cotas two sets of three ters, wth opposte sgs, as the secod set correspods to a double exchage. By aalogy wth Eq.(10), ths leads us to wrte the coutator (8) as [D, F ] 1 = F ρ { ( ) } λρ ( ) + D, (11) where the operator D = F D (1) gves zero whe actg o vacuu. () To get rd of ths operator D, we eed a thrd coutato relato. The choce betwee coutator ad atcoutator s aga ade by eforcg a (, k) syetry [ [[F, F ] +1, F ] ] [ [[F, F 1 k =, F ] +1, F ] ] k, F η η (12) Ths requres η 3 = +1. After soe algebra, we ed wth [D, F k ] +1 = p k F pf χ + per. (13) p, p k where the scalar χ correspods to the Kal dagra show Fg.3, wth all possble perutatos of the dces o the rght, aely, the crcular perutatos whch trasfor (,, k) to (, k, ) ad (k,, ), ad the three o-crcular perutatos whch trasfor (,, k) to (, k, ), (,, k) ad (k,, ), the last three ters appearg wth a opposte sg.

8 8 2.2 Fero teractos betwee coposte partcles () We ow tur to scattergs duced by fero teractos. I order to choose betwee coutator ad atcoutator, we ca ote that H actg o a state ade of oe of these coposte partcles plus a arbtrary state ψ ust gve HC ψ = E C ψ + C H ψ +..., (14) for (H E )C v = 0. Ths leads us to cosder the coutator of H ad C for both, cobosos ad coferos. Ths coutator, wrtte as [8,10] allows to defe the creato potetal V [H, C ] 1 = E C + V, (15) of partcle. It descrbes the teractos of ths partcle wth the rest of the syste, due to the eleetary teractos of ts feroc copoets. Fro hoogeety, ths operator ust read as V = C V (1), wth V (1) v = 0, as obtaed fro V v = 0, whch readly follows fro Eq.(15) actg o vacuu. A precse calculato of ths operator shows that t reads as a su of products lke a a. () To get rd of the operator V, we eed a last coutato relato. The choce betwee coutator ad atcoutator aga follows fro (, ) syetry [ [H, C ] 1, C ] [ = η η 2 1 [H, C ] η1 η 2, ] C. (16) η 1 Ths requres η 2 = η 1,.e., a coutator for cobosos ad a atcoutator for coferos. Hoogeety the leads to wrte [8,10] [V, C ] η1 = C C ξ ( ), (17), where ξ ( ) s a scalar. Ths scatterg correspods to teractos betwee the eleetary feros of the coposte partcles (, ), the absece of fero exchage,.e., wth ad ade wth the sae feros. It s represeted by the dagras of Fg Structure of the key equatos We see that the scatterg betwee two coposte quatu partcles, whch coes fro eleetary-fero teractos, follows fro a frst coutator betwee the syste

9 9 Haltoa ad the creato operator of the partcle at had, whatever the partcle s, boso-lke or fero-lke. Such a coutator geerates a creato potetal whch descrbes the teracto of ths coposte partcle wth the rest of the syste. By takg the coutator or the atcoutator of ths creato potetal wth a secod partcle creato operator the choce depedg whether the partcles are boso-lke or fero-lke we ca reach the drect scatterg of the two coposte partcles cog fro teractos betwee ther eleetary feros, the absece of fero exchage. The scattergs cog fro fero exchages the absece of fero teracto are ore subtle to geerate. They coe fro two coutators the case of cobosos ade of 2 feros. For coferos ade of 3 feros, we eed two atcoutators plus oe coutator, whle for 4-fero partcles, we eed four coutators, ad so o...ths set of coutato relatos allows us to geerate all fero exchages takg place the scalar products of two-partcle states, three-partcle states, etc... These exchages are vsualzed by ultar dagras that we have called Shva ad Kal, the correspodg exchage scattergs beg readly calculated fro these dagraatc represetatos, through fully tutve rules. 3 A few sple applcatos Sce exchages betwee two-fero partcles were extesvely studed our prevous works o coposte exctos [10], let us ed ths letter by a few probles volvg exchage processes betwee coferos ade of three feros. () Accordg to Eqs.(6,11), the scalar product of two-cofero states s gve by { v F F F F v = δ, δ, ( )} λ ρ { }, (18) ρ where λ ρ ( ) correspods to the Shva-lke dagra of Fg.2(a) whch the two coferos exchage a fero ρ. We readly recover that ths scalar product reduces to zero for ( = ) or ( = ), as ecessary sce F 2 v = 0, due to Eq.(5). The above equato also shows that the oralzato factor of a two-cofero state F F v, wth v F F v = 1 for = (, ), s gve by F F F F v = (1 δ, ) [ 1 ρ { ( ) ( )}] λρ λρ = (1 δ, )[1 X ]. (19)

10 10 As for eleetary feros, ths oralzato factor reduces to zero for =. However, ulke the, t s ot exactly equal to 1 for, due to the Paul excluso prcple whch geerates the exchage ter of Eq.(19). Such a oth-eate effect, resultg fro ths Paul excluso whch eforces the secod cofero to be coplete, s expected to decrease the eleetary fero value of the scalar product, the sae way as for coboso states. Ideed, as prevously show [11,12], the oralzato factor for N detcal cobosos 0, aely, v B N 0 B N 0 v, reads as N! F N, where F N, always saller tha 1, turs expoetally sall the large N lt. () A slar oth-eate effect also exsts for the oralzato factor of threecofero states. By usg Eqs.(6,11,13), we fd that, for dfferet (,, k), ths oralzato factor reads v F k F F F F F k v = 1 (X + X k + X k ) + S 2 + S 3 + K. (20) The frst ter, 1, coes fro process whch the three coferos keep ther three feros, as f these were eleetary partcles. I the secod ter, two coferos aog three are volved exchages slar to the oes appearg Eq.(19), the thrd cofero stayg uchaged. The ter S 2 correspods to the Shva-lke dagra of Fg.2(d), wth two coferos aog three explodg to (2+1) feros. The ter S 3 correspods to the Shva-lke dagra of Fg.2(c) whch all three coferos explode to (2+1) feros. Fally, the last ter K correspods to the Kal dagra of Fg.3(a) or Fg.3(b), whch all three coferos explode to (1+1+1) feros. Ths scalar product s deftely rather awful. We ust however ote that we actually are hadlg 3 3 feros o each sde,.e., 18 quatu partcles. Thaks to the Shva ad Kal dagras troduced to vsualze coposte-partcle ay-body effects, we ca ot oly uderstad these exchages but also classfy the a systeatc way, a bld brute force calculato, always possble whe the uber of coferos s sall, beg hardly extedable to larger uber of coferos. () We ca also study the eergy of cofero states through the Haltoa ea value, as we dd for cobosos [10,13]. I the two-cofero subspace, ths Haltoa ea value reads as v F F HF F v v F F F F = [E + E ] + C. (21) v

11 11 The frst ter correspods to the eerges of the free coferos ad, whle the secod ter coes fro teractos, ts precse value readg C = [ ξ ( ) ( )] [ ( ) ξ ( )] ξ ξ 1 ρ [ ( ) ( λρ λρ )]. (22) The deoator coes fro the oralzato factor whch s ot exactly 1 due to fero exchages betwee coferos. The uerator cotas drect ad exchage processes slar to the oes we foud for cobosos [10]. Note that the feroc ature of the partcles, whch leads to F F v = F F v, duces a us sg the ( ) perutato, whch does ot exst the case of cobosos. The scatterg ξ ( correspods to the drect teracto process appearg Eq.(17) ad represeted Fg.6(b), the coferos ad keepg ther three feros. By cotrast, the scatterg ξ ( ) correspods to a exchage teracto process, coferos ad exchagg ust oe fero sce a two-fero exchage s equvalet to a oe-fero exchage wth a dex perutato. Ths exchage scatterg s defed the sae way as for cobosos [10], aely, ξ ( ) = p,q ρ ( ) ( λ q ρ p ξ q p ) ; (23) ) 4 Cocluso Through a rather tutve aalyss of the fero exchages takg place betwee coposte quatu partcles, we have detfed the possble topologes of the dagras represetg these exchages. I addto to dagras slar to the Shva dagras troduced to vsualze the ay-body physcs of cobosos ade of two feros, we here show that a ew set of three-ar dagras, called Kal, s ecessary to properly represet fero exchages takg place betwee coferos ade of three feros. We also show how these fero exchages ca be geerated fro hard algebra, through a set of coutators ad atcoutators slar to the oes we troduced the ay-body theory of cobosos ade of two feros. The coplexty of these exchages creasg rapdly wth the uber of feros cotaed these coposte obects, ther vsualzato through Shva ad Kal dagras, wll appear as hghly valuable to cotrol the ay-body physcs of such coposte quatu partcles.

12 12 We wsh to thak Marc-Adré Dupertus for valuable dscussos at the begg of ths work. Refereces 1 A. Fetter, J. Walecka, Quatu Theory of May-Partcle Systes (McGraw-Hll, New York, 1971). 2 G.D. Maha, May Partcle Physcs (Pleu, New York, 1981). 3 For a revew, see for exaple, A. Kle, E.R. Marshalek, Rev. Mod. Phys. 63, 375 (1991). 4 M.D. Grardeau, J. Math. Phys. 16, 1901 (1975); C. Lo, M.D. Grardeau, Phys. Rev. A 41, 158 (1990). 5 M. Cobescot, Eur. Phys. J. B 60, 289 (2007). 6 E. Haaura, H. Haug, Phys. Reports C 33, 209 (1977). 7 H. Haug, S. Schtt-Rk, Prog. Quatu Electro. 9, 3 (1984). 8 M. Cobescot, O. Betbeder-Matbet, Europhys. Lett. 58, 87 (2002). 9 M. Cobescot, O. Betbeder-Matbet, Eur. Phys. J. B 55, 63 (2007). 10 For a revew, see M. Cobescot, O. Betbeder-Matbet, F. Dub, Phys. Reports 463, 215 (2008). 11 M. Cobescot, C. Taguy, Europhys. Lett. 55, 390 (2001). 12 M. Cobescot, X. Leyroas, C. Taguy, Europhys. Lett. 31, 17 (2003). 13 O. Betbeder-Matbet, M. Cobescot, Eur. Phys. J. B 31, 517 (2003).

13 13 (a) p k p k k p (b) (d) (e) Fgure 1: Usual star-topology (c) for Shva(f) dagras represetg fero exchages betwee two (a), three (b), ad N cobosos (c) ade of two feros α (sold les) ad β (dashed les). These exchages ca also be represeted through le-topology as show (d,e) or (f). = (a) (b) p k p k Fgure 2: Shva-lke dagras for(c) fero exchages (d) betwee coferos ade of three feros (α, β, γ) represeted by sold, dashed ad dotted les. I (a,b), the two coferos explode to (2+1) feros. I (c), the three coferos explode to (2+1) feros, whle (d), oly two coferos explode to (2+1) feros.

14 14 k p (a) p k Fgure 3: Kal dagras for (b) fero exchages betwee three coferos whch explode to (1+1+1) feros. The sae exchage process ca be equvaletly represeted a star-topology (a) or a le topology (b). We have also represeted the (α, β, γ) les separately, to ake the dagra topologes clearer. Fgure 4: (a,b,c) Shva-lke dagras for fero exchages betwee (a) (b) (c) (d) four coferos whch four, three, two coferos explode to (2+1) feros. (d) Mxture of Shva-Kal dagra whe oly oe of the four coferos explodes to (2+1) feros.

15 15 (a) (b) Fgure 5: The two possble Kal dagras a le-topology for fero exchages betwee four coferos explodg to (1+1+1) feros. The (α, β, γ) les, wrtte separately, help to see the topology of these two dfferet dagras. = (a) (b) Fgure 6: (a) Drect scatterg betwee two cobosos ade of two feros, resultg fro teractos betwee ther feroc copoets. (b) Sae for two coferos ade of three feros.