imr CDT Lecture 07: Groups and Algebras

Transcription

1 imr CDT Lecture 07: Groups d Algebrs Before we proceed to itroduce groups d lgebrs, it is useful to review the digrm from the previous lecture d ote tht we hve ow fully explored the prt delig with spces d Hilbert spces. Rigorous defiitios re give below, but for overview purposes the reltioship betwee spces, groups d lgebrs c be illustrted o spi opertors: Spce: Group: opertors d their lier combitios. orm d ier product optiolly defied. o products betwee opertors or iverse opertors. opertors, their products d iverses. All opertors must be ivertible. o lier combitios, orms, ier products or multiplictio by sclrs. Algebr: opertors, their lier combitios, products d fuctios. orm d sclr product optiolly defied. The set of qutum mechicl opertors, s covetiolly viewed by physicists, is therefore lgebr cosistig of spce of opertors with their superpositio used s product. Certi sets of opertors (otbly mtrix expoetils) re groups uder tht product. The sclrs come from or. Semigroups d groups A set G equipped with ssocitive biry product *:GG G is clled semigroup if it is closed uder tht biry product: gh, G gh G () If dditiolly the followig reltios hold i G :.! eg: gg eg g (there exists uique uit elemet). g G! g G: gg e (for every elemet there exists uique iverse) The the set is clled group. The elemets of G eed ot commute, but if they do, the group is clled Abeli. The umber of elemets i the group is clled the order of the group. A subset of G which is itself group is clled subgroup of G. Two groups G d H re clled isomorphous if there exists oe to oe correspodece :G H g, g G, g g g g. such tht

2 Exmples of groups commoly ecoutered i spi dymics iclude:. The set of ll time propgtors exp( iht ) uder sttic Hmiltoi Ĥ. It is clled the propgtor group of Ĥ. The properties listed bove re esy to demostrte product of y two propgtors is lso propgtor: there is uit propgtor tht does othig: iht iht iht iht ih tt e, e G e e e G () iht! : iht iht EG e G Ee e (3) Ad for every propgtor there is uique iverse tht tes the system bc i time: iht iht iht iht iht e G! e e G: e e E (4) Becuse the Hmiltoi commutes with itself, this is Abeli group. If relxtio is preset, the propgtors do ot, i geerl, hve uique iverse the set of ll time propgtors uder Liouvilli tht icludes relxtio is therefore semigroup.. The set of ll possible permuttios of severl ideticl spis i spi system. This group is ot Abeli becuse the order of permuttios does ffect the result. The permuttio group of objects is usully deoted S. 3. Symmetry groups ecoutered i electroic structure theory d crystllogrphy. All fiite poit groups describig moleculr symmetry re subgroups of S for some. 4. Automorphism groups of Hilbert spces. It is esy to see tht the set of ll ivertible lier opertors ctig o the wvefuctio spce of physicl system is group. The defiitios of group d semigroup give bove re too geerl for our purposes, d i the tretmet below we shll therefore wor specificlly with groups of qutum mechicl opertors d superopertors, which will iherit the orm d ier product from the correspodig spce. The brodest of such groups the group mde by the set of ll ivertible mtrices over field F together with mtrix multiplictio is clled the geerl lier group GL, F. I the tretmet below we shll lso ssume F or F. Discrete d fiite groups For give elemet gg GL d y positive, the set of ll opertors h such tht g h (5) is clled the eighbourhood of g. If every elemet of the group G hs eighbourhood i GL tht does ot coti other elemets of G, the group G is clled discrete. I other words, GL seprtely d c be eumerted. If group the opertors of discrete group re plced i hs fiite umber of elemets, it is clled fiite group. All other groups re ifiite. Exmples of fiite groups iclude permuttio groups d poit groups describig moleculr symmetry. Crystllogrphic symmetry groups re discrete, but ifiite, becuse they coti ifiite umber of distict crystl cell trsltio symmetry opertios.

3 Cotiuous groups A group G is clled cotiuous or Lie group if it is lso differetible mifold, tht is, if ech opertor g G hs eighbourhood with the followig property: g t t of vribles, which is cotiuous withi the ti cube, defies ll opertors withi the eighbourhood of g d hs differet vlues for differet prmeters t,..., t. There exists opertor vlued fuctio,..., Such fuctio is clled the prmeteriztio of the group roud g. It c be show tht the umber of vribles is uiquely determied by the group. It is clled the dimesio of the group, d the group itself is clled prmetric. I spi dymics, Lie groups mostly rise i the cotext of three dimesiol rottios d time evolutio. We will ecouter the followig Lie groups: Geerl lier group, GL of ivertible mtrices. Specil lier group, SL of ivertible mtrices with det. Uitry group, U of uitry mtrices. Orthogol group, O of orthogol mtrices. Specil uitry group, SU of uitry mtrices with det. Specil orthogol group, SO of orthogol mtrices with det. SU is the group of trsformtios of the wvefuctio spce of sigle spi / pr I prticulr, ticle d SO is the group of three dimesiol rottios i rel spce. 3 Group ctios, orbits d stbilizers The spi system trjectory uder give Hmiltoi: t iht e 0 (6) shres the prmeter (time) with the propgtor group. The system trjectory my therefore be viewed s sequetil pplictio of every elemet of the propgtor group to the iitil stte of the system. More formlly, if G is group of opertors ctig o set A, the the group ctio by G o A is mp G: A A, tht is, g, where A, g G. I o Abeli groups, left d right ctio my be defied i the obvious wy. A subset B of the set A is clled ivrit uder the ctio by G if gb bb, gg B d fixed uder G if gb b for ll g G d b B. If G is group of opertors ctig o set A, the the group orbit of elemet A is the subset of A to which my be moved by the ctio of elemets of G : G g g G (7) t is the propgtor group orbit of the iitil stte Therefore, the spi system trjectory 0. The orbits re equivlece clsses o A uder the followig equivlece reltio: x y if gg: gx y (8)

4 The set of orbits of ll elemets of A forms prtitio of A, tht is, distict orbits ever itersect. If A is spce, the differet orbits belog to differet subspces of A. The set of ll orbits of A uder the ctio of group G is clled the quotiet or the orbit spce of the ctio d deoted A / G. The subset of G levig prticulr poit A ivrit is subgroup of G. It is clled the stbilizer subgroup G of : G gg g (9) If ll stbilizers re trivil, the group ctio G: A A is clled free. I the spi dymics cotext the ctio by the propgtor group is rrely free due to the existece of the coservtio lws. Cojugcy clsses d cetres Two elemets b, Gre clled cojugte if gg, gg b. The set of ll elemets of the form gg with g G is clled the cojugcy clss of : Cl gg g G (0) Ech elemet of group c belog to just oe cojugcy clss. Two cojugcy clsses of group re either ideticl, or disjoit the group is prtitioed ito cojugcy clsses. It is obvious tht the idetity elemet lwys forms its ow clss d the umber of clsses i Abeli groups is equl to the order of the group. The umber of elemets i ech cojugcy clss is divisor of the group order d the umber of uique (up to similrity trsformtio) irreducible represettios of group is equl to the umber of cojugcy clsses. The cetre G of group G is the set of elemets tht commute with every elemet of G : G z G g G, zg gz () It is esy to see tht the cetre is Abeli subgroup of G. A group with oly the idetity elemet i the cetre is clled cetreless. Algebrs A vector spce A over field F equipped with biry product *: A A A is clled lgebr, if:. b, A b A (the spce is closed uder the product) (the biry product is left distributive). bc,, A bc c bc 3. bc,, A b c b c 4. b, A,, F b b (the biry product is right distributive) (multiplictio by sclr is ssocitive) ote tht there is o requiremet for the biry product to be ssocitive d o requiremet for uit elemet or uique iverse to exist. The set ll of spi opertors, log with their superpositios d lier combitios, is therefore lgebr. Sublgebrs re defied i the usul wy. A lgebr iherits bsis sets (d the metric, if it exists) from the pret spce. Ech elemet of lgebr thus hs uique expsio vi the elemets of the chose bsis set: () e b e where e is the bsis set. I prticulr, this is true for products of bsis elemets:

5 ee cmem (3) m Therefore, the expsio coefficiets for the product of y two elemets: (4) b ee c e e, c m m m m m m m m re completely determied by the expsio coefficiets of the operds d the rry of structure coefficiets c m. The structure coefficiets uiquely defie the lgebr. Bsis sets with simple structure reltios (for exmple, Gussi fuctios d sphericl hrmoics i Qutum Chemistry) re vluble becuse they fcilitte computtiolly efficiet implemettios of my prcticl opertios. Lie lgebrs A vector spce over field of complex umbers (this is ot strictly ecessry, but is dictted by physics) equipped with biry opertio, : ( Lie brcet), is clled Lie lgebr, if:. [ b, c] [, c] [ b, c] bc,,, (the brcet is bilier) [, bc] [, b] [, c]. b, [ b, ] [ b, ] (the brcet is tisymmetric) 3., b, c [[, b], c] [[ b, c], ] [[ c, ], b] 0 (Jcobi idetity holds) 4. (mthemtics) b, b, (spce is closed uder, ) 4. (physics) b, ib, (spce is closed uder, i ) I the cotext of qutum mechics it is ofte the cse tht the spce i questio is of Hermiti opertors, which correspod to physicl observbles. It is esy to see, however, tht the commuttor of two Hermiti opertors [,] b is ti Hermiti. This is icoveiet d for this reso the defiitio commoly used i physicl scieces icludes the i term i Coditio 4 writig the commuttio opertio s [,] b icesures tht ĉ is lso Hermiti opertor. The coveiece of the physicists formultio is prticulrly evidet from Liouville vo eum s equtio. Two Lie lgebrs d over field F re clled isomorphous, if there exists oe to oe mp :, such tht. b, b b. F 3. b, b,, b A fiite set of opertors { e,..., e } from Lie lgebr is clled bsis, if every opertor i c be represeted s lier combitio of { e,..., e } : e (5) Importtly, whe lgebr is defied over specific field of sclrs, it is the coefficiets of such lier combitios tht come from tht field, but ot ecessrily the mtrices. If the bsis is lierly idepedet the expsio bove is uique. For commuttor of two bsis set elemets, we get:

6 e, e i c e (6) m m The coefficiets c m re ow s the structure costts of the Lie lgebr. They re determied by the choice of the bsis set. I the prcticl simultio cotext, the structure costts c ofte be obtied lyticlly. They llow the clcultio of commuttors without recourse to y specific represettios: b, e, e i c e m m m m (7) m From the defiitio of the commuttor d from Jcobi idetity it is esy to demostrte tht: cm cm, c 0, cmpcpq cmpcpq cpcpmq 0 q (8) Commoly ecoutered Lie lgebrs iclude: p. gl the Lie lgebr of ll lier opertors. The most populr bsis set is composed of mtrices ij ij b with sigle uit elemet i i th row d j th colum: b pq piqj. Therefore, for the commuttor of two such mtrices: ij ij ij ij ij [ b, b ] b b b b b b b b r pq pq r pr rq r pr rq ip jr r q i j p r ir jq ip j q p i jq j b i b pq r i lm l im [ b, b ] imb lb (9). sl the Lie lgebr of ll trceless lier opertors. Oe of the most coveiet bsis sets is Oubo mtrices, for which: i i b i b... b (0) i lm rs [, ] rs m im lr s l ir sm 3. u the Lie lgebr of ll Hermiti opertors. We could choose the bsis mtrices to be Hermiti lier combitios of oe elemet mtrices b ij : b, b b, i b b () ij but it is ofte coveiet to use exterl bsis set, such s b themselves, which does belog to lrger lgebr, but still geertes u becuse u gl. Coveiet iterl geertors for u, origilly proposed by Wolfgg Puli, re i 0 e,,, 3 () 0 0 i su the Lie lgebr of ll trceless Hermiti opertors. Oe of the possible iterl geertor sets my be obtied by formig Hermiti lier combitios of Oubo mtrices:,, i (3) I the cse of, if mtrix is trceless, the uit mtrix Ê does ot pper i its expsio. Therefore, the,, 3 Puli mtrices re iterl geertors of su. pq

7 Expoetil mp Let us lyse the properties of expoetils of the mtrices belogig to oe of the lgebrs described bove. It is esy to prove the followig sttemets: Tr. det e e. For y mtrix, 3. If [ b, ] 0, the e is ivertible opertor d its iverse is b b e e e. 4. For y opertor gl, e e. i i 5. If is Hermiti, the e is uitry, d if is trceless, the det e. It therefore ppers tht the expoetils of the mtrices ihbitig the lgebrs listed bove stisfy the defiitio of Lie group they re ivertible d cotiuous with respect to the coefficiets of the lier combitios i the pret lgebr. More specificlly, we hve see bove tht Lie lgebr c be prmeterized by set of bsis opertors e d coefficiets d the followig reltio provides mp betwee the group d the lgebr: e.,..., e,,..., g,..., expiaexp ie i e G! I the physics cotext, the most commo form of mtrix expoetil is expi. Uder this defiitio of the expoetil mp, the followig correspodece holds (the fields refer to lier combitio coefficiets iside the lgebrs tht prmeterize the correspodig groups): Lie group Dimesio GL, GL, Mtrices i the group Lie lgebr ivertible rel, ivertible SL, ivertible, det SL, SO, U, / SU, rel, ivertible, det rel, ivertible, orthogol Mtrices i the lgebr gl ll gl imgiry sl trceless sl so ivertible, uitry ivertible, uitry, det imgiry, trceless imgiry, Hermiti u Hermiti su Hermiti, trceless (4) Adjoit ctio I the sme wy s the group opertio defies ctio of the group o itself, the Lie brcet my be used to defie ctio by lgebr o itself. Give elemet of Lie lgebr, the djoit c

8 tio of o is edomorphism d : d b [, b]. I spi dymics lguge, the djoit ctio by opertor â correspods to the ctio by the commuttio superopertor iduced by â : d [ b b, b]. with Adjoit ctio defies represettio of Lie lgebr (it is clled the djoit represettio) becuse the djoit ctios stisfy the sme commuttio reltios s the origil opertors: d,d d [, ] [, ],, b c b c L S M M L S (5) It is esy to see from the defiitio tht the djoit represettio correspods to the trsformtio betwee Hilbert d Liouville spce formlisms i spi dymics: t i[ H t, t ] i H t t H t d H t [ H t, ] t (6) exp iht exp iht exp iht The djoit represettio ( Liouville spce) is very coveiet becuse time propgtio is ccomplished by oe sided multiplictio opertio o colum vector, rther th two sided multiplictio o mtrix this fcilittes my lyticl clcultios. Mtrix represettios of groups d lgebrs Becuse clcultios re typiclly performed o digitl computers, it is i prctice coveiet to wor with mtrix represettios of opertors (rther th, for exmple, their differetil forms). A P: G GL, such tht: dimesiol mtrix represettio of group G is mp gh, G P gh P g Ph (7) d uit opertor i G is represeted by uit mtrix i GL. Some loss of iformtio my occur uder this mp. I this regrd, group represettios c be: Fithful: whe P is isomorphism, tht is, differet elemets of G re mpped ito differet elemets of GL. A commo exmple of fithful represettio is the oe commoly used i qutum mechics: if G is group of opertors o Hilbert spce d is orthoorml bsis of tht spce, the the set of mtrices with the followig elemets: g P g (8) where g G, is fithful represettio of the group G. Regulr: fithful represettio tht uses the group itself s bsis (this is possible becuse G i i is ble to ct o itself by multiplictio). If we defie j such tht gg i j jg, the the mtrix correspodig to the regulr represettio of g i i is P gi j j. The dimesio of the regulr represettio is equl to the order of the group. Lossy: represettio i which two or more elemets of G re mpped ito the sme elemet of GL. Some properties d reltios betwee the elemets of the group my survive, others would be lost.

9 Sclr: lossy represettio i which ech g G is mpped ito the determit of its fithful represettio. This represettio is lwys Abeli. Trivil: lossy represettio i which ll elemets of G re mpped ito the uit mtrix. Two represettios P d Q of the sme group G re clled equivlet, if they hve the sme dimesio d there is similrity trsformtio tig oe ito the other: gg Q g W P g W (9) All represettios of fiite groups re equivlet to uitry represettios. Let P be dimesiol mtrix represettio of G d let the represettio spce hve S decompositio ito direct sum of subspces..., such tht the idividul subspces i re ivrit uder the opertors represetig the elemets of G. The the represettio P P... PS is direct sum of represettios P,..., P S. If bsis is selected i ech of the subspces d their direct sum is used s the bsis of, the mtrices of P cquire bloc digol i form, with idividul blocs correspodig to idividul represettios i the direct sum. Let P d Q be represettios of G i P g Q g defied o of the form d K respectively. The Kroecer products of mtrices K re other represettio of G, becuse PgQgPhQh P gh Q gh P g P h Q g Q h this represettio is clled direct product of represettios P d Q. A mtrix represettio P of group G i GL is sid to be reducible if it c be cst by some similrity trsformtio ito the followig form: P g X g 0 P g P g gg where P g, P g d X g re blocs of the mtrix P g. A represettio is fully reducible if it c be trsformed ito direct sum of represettios of lower dimesio: P g 0 0 Pg 0 Pg 0 PgPg 0 0 If does ot coti y subspces other th itself tht re ivrit with respect to ll the opertors i P, the represettio P is clled irreducible. Ay represettio of fiite group is either irreducible or fully reducible. For ifiite groups, if represettio is uitry d reducible, the it is fully reducible. A very geerl orthogolity reltio (ow s the gret orthogolity theorem) exists betwee two uitry irreducible mtrix represettios P d P b of group G : G b (30) (3) (3) * Pg Pbg b jlm (33) gg j lm

10 where is the order of the group d, b re dimesios of the two represettios. I other words, if the represettio mtrices re stced o top of oe other lie dec of crds o tble, the colums of the two stcs re orthogol for differet represettios. Chrcters The chrcter of mtrix represettio P of group elemet is the trce of the correspodig mtrix: P g Tr P g P g (34) The chrcter of represettio is ordered set of chrcters of the idividul elemets of the group (usully writte s row vector). Trce is ivrit uder similrity trsformtio: Tr W P g W Tr WW P g Tr P g (35) meig tht ll equivlet represettios hve the sme chrcters. From the defiitio of cojugcy clss it lso follows tht ll members of cojugcy clss hve the sme chrcter. After computig the trces of the mtrices o the left hd side of the gret orthogolity theorem, we get: b * P g P g j b lm j lm b jl m j lm jlm G gg jlm b * P g P g b mm b m m G gg m * P g P g b mm b G gg m G gg g g * P Pb b Tht is, chrcters of differet irreducible represettios re orthogol s vectors (this is ow s the little orthogolity theorem). A chrcter tble is tble of chrcters for ll uique (up to similrity trsformtio) irreducible represettios of the group. For the commo exmple of S 3 group ( C 3v the symmetry group of H 3 molecule) we hve: (36) E C 3 (0 ) C 3 (40 ) σ σ σ 3 A A E It is esy to see tht the lies of this tble re orthogol s vectors. Chrcter tbles for ll commoly ecoutered groups re tbulted i the literture.

11 Applictios of fiite groups to spi dymics Chrcter tbles re useful for the costructio of symmetry dpted lier combitios (SALCs) of bsis fuctios or opertors. SALCs sp subspces tht re ivrit uder the ctio of the opertors of the group. Becuse the Hmiltoi commutes with the system symmetry opertors, the sme subspces re lso ivrit uder the Hmiltoi, which is therefore bloc digol i the SALC bsis. Hmiltoi bloc digoliztio c sve sigifict computtiol resources i prcticl clcultios ech bloc c be simulted seprtely d it is ofte the cse tht some blocs re upopulted i the iitil coditio d c therefore be dropped ltogether. Figure. o zero ptter of the Hmiltoi commuttio superopertor of rdicl pir with four equivlet protos before (left) d fter (right) symmetristio of the bsis set uder the lrgest Abeli subgroup of S 4. Oly oe of the resultig blocs is populted i typicl simultios. The symmetry blocs correspod to coservtio lws withi the physicl system. A simultio istce tht strted off i specific ivrit subspce of the symmetry opertor would sty there idefiitely. As the commo lbortory jrgo puts it, the sttes of differet symmetry do ot iterct. For discrete groups SALCs re give by the followig equtios: g O g O (37) g g gg gg where is the ormliztio costt, the summtio is crried over the idividul elemets g of the symmetry group G, g is the chrcter of irreducible represettio of the group elemet g, d go ( ) is the result of the ctio by tht group elemet o the bsis opertor O. For spi opertors, the group ctio g:{ O } { O} is edomorphism of the system stte spce tht mouts to permuttio of the order of the direct product compoets of { O }. A exmple is give below for the group ctio tble of C 3v o proto spi opertors of H 3 : Opertor E C 3 (0 ) C 3 (40 ) σ σ σ 3 EE E EE E EE E E E E E E E EE E E E E L E E L E E EL E E E L L E E EE L E L E E L E E E L E E L EL E EE L L E E E E L EL E L E E L E E E L E E L E L E E E E L

12 Liouville spce symmetry tretmet I my prcticl cses, oly oe irreducible represettio eeds to be simulted i Liouville spce the oe with ll chrcters equl to uity, otherwise ow s A g, otherwise ow s g, or the fully symmetric irreducible represettio. This differece i symmetry behvior of Hilbert d Liouville spces my be illustrted o two spi system while the siglet stte wvefuctio does chge sig uder spi permuttio i Hilbert spce: P (38) its represettio i Liouville spce does ot: P P (39) More geerlly, y symmetry dpted Hilbert spce wvefuctio belogig to irrep other th the fully symmetric oe i P e, (40) is goig to hve fully symmetric represettio i Liouville spce becuse i i P P e e (4) Therefore irreducible represettios other th A g get symmetrized ito A g durig the Hilbert to Liouville spce trsformtio. It lso ofte the cse tht system evolutio i Liouville spce stys cofied to. I the most geerl cse, the evolutio is govered by the followig equtio: A g ih K R eq (4) t where is the desity mtrix, eq is the equilibrium desity mtrix, Ĥ is the Hmiltoi commuttio superopertor, K is the chemicl ietics d sptil diffusio superopertor d R is the relxtio superopertor obtied usig oe of the vilble relxtio theories. The Hmiltoi commuttio superopertor H [ H, ] E H H E, where Ê is the idetity opertor, iherits the T symmetry of the Hmiltoi which commutes with the symmetry group, meig tht H Ag. If the user declred some spis equivlet i system udergoig chemicl rectios, they must be trsported s such i y chemicl process, meig tht, by defiitio, K Ag. The iitil stte of the system is either thermodymic equilibrium, which iherits the A symmetry of the Hmiltoi vi eq exp HT B Tr exp HT B or user supplied o equilibrium stte, which is fully symmetric by defiitio with respect to the spis tht the user declred equivlet. The cse of the relxtio superopertor is somewht more ivolved. I the cse of the Redfield superopertor 0 ih0 eq eq 0 ih Uder the ssumptio tht the stochstic prt R [ 0,[ H e H e, ]] d (44) Ĥ the etire double commuttio superopertor o the right hd side iherits the symmetry of Ĥ 0 d Ĥ t, which is A g. otig tht the direct product of y umber of fully symmetric irreps is itself g (43) t of the Hmiltoi obeys the system symmetry,

13 fully symmetric irrep completes the proof. I summry, i symmetric systems with symmetric iitil coditios the symmetry would ot be broe spoteously becuse ll evets d opertors ffectig it belog to A g. Symmetry tretmet for MR d ESR systems For y trsformtio to be symmetry, it is requiremet tht the correspodig superopertor leves the Hmiltoi uchged. Exmples iclude lbel permuttios o ideticl spis: 3 3 H S S S JL S S S 3 3 S S S JL S S S but oly i the cse where the couplig structure (icludig tesor eigevlues d eigevectors) to y exterl spis is similrly ivrit. This is ofte the cse i liquid stte ESR spectroscopy, where ll uclei with similr hyperfie couplig costts c be declred equivlet uder the full permuttio group. The sme pplies to methyl d isopropyl groups i liquid stte MR. Situtios where the Hmiltoi is ivrit uder simulteous lbel permuttios of groups of spis tht re ot relted by simple permuttio symmetry must be treted o cse by cse bsis. I prticulr, i solid stte simultios the permuttio symmetry c be pplied if d oly if the uclei hve ideticl iterctio tesors to the sme prter spis. It must be demostrted tht the Hmiltoi is ideed ivrit uder ll group opertios, the group ctio tble must be built for ech elemet of the bsis set d multiplied ito the chrcter tble to get the symmetry dpted bsis. (45)