A Fermionic ITO Product Formula

Transcription

1 IJIET - Ieiol Joul of Iovive ciece Egieeig & Techolog Vol. Iue 3 Mch 5..ijie.com A Femioic ITO Poduc Fomul Cii Şeăecu Uivei Poliehic Buche Deme of Mhemic Buche 64 Romi Ac We ove Io oduc fomul fo ochic iegl o Femio Fock ce log ih he me kid of iegl o Boo Fock ce. Becue of he ocommuivi elio he oie heo of ochic iegio mu diiguih eee he lef d igh iegl. We hve fou uch oiiliie. Keod: ochic iegl Femio Fock cec*- lgeie oduc.. Ioducio Fi e couc ochic iegl o Femio Fock ce [7][] log ih he me kid of iegl o Boo Fock ce [4] fi of imle ocee. We defie he Femio ochic iegl fo que-iegle iegd. We ee he ifiieiml oeo like ifiie um u e ume he e coiuou. Becue of he coicl icommuio elio e hve lef igh d mi ochic iegl[]. We ecll givig ol he ece deil ome coce d oio. Defiiio.. e FGH e imle ocee d ie F F H H G G The fmil of oeo M M defied M M A A F D M h h ih M G A A H fo i clled igh ochic iegl of F GH d i deoed : M da F GdA Hd imill e c defie he lef ochic iegl of F GH deoed N N A A F A A G H Fo N da F da GHd Ad he mi ochic iegl of FGH deoed : P FdA da G Hd P da F GdA Hd Theoem.. e FGH e imle ocee d e ume h F h i hich me h ll F hve he i d le e dm daf GdA Hd If uv h H H o 83

2 J k IJIET - Ieiol Joul of Iovive ciece Egieeig & Techolog Vol. Iue 3 Mch 5..ijie.com j J j The: M u v M u v i oeo h i he elf djoi ui oeo ich i o h d o h. A oeo T : k h h i id o e eve if k F Fu vck T k T d odd if T T d T h he i u if G v J k j J j Cj j G v H u v Cj j We geelize he elio of heoem.. : v v' v' ' ' v' v hece e elce k ih ih k d ih u H d v. Poof: ee i [3].. Oeo i Defiiio.. We ume h h i Z gded ih eve d odd uce h h d e deoe he T e H e Hile ce. We defie he immeic Hu v Fock ce H ove H he lie hull of ll ih (hee fo e hve he ui eleme mel ) ih he folloig Poof: ee i [3]. ie oduc: k Theoem.3. e FGH e imle ocee k de i i dm daf GdA Hd i j Fo u h H fo k he deemi i coideed o e ). d v h Aou hi ce e meio he folloig: H i. H H The: hee H i he cloed lie hull of ll M M u v ih k F Fu vck ii. k hee i o - if i eve o odd. T If o i ih diffee idee i e equl hi oduc i ull. iii. H H fiie i ociive lge ih ui eleme d i he oduc of ih H i elihed ode. iv. If H K he H K. e e H H H H H eve H H odd We deoe he i oeo o H I I d 84

3 IJIET - Ieiol Joul of Iovive ciece Egieeig & Techolog Vol. Iue 3 Mch 5..ijie.com Av d Remk The oeo A v hve he i o h. We c ie: A v A v Ad if hh h h H h e c decomoe A A I A e j eecivel A A I A e j 3. A ITO Poduc Fomul Theoem 3.. e FGH e imle ocee d dm da GdA Hd. We ume h ll hve J J k M he me i. Fo uv h ih ih e hve: M u M v M u M v F u F v k k M u v u M v d k k hee P d ik ill e ecificed i he folloig. Poof: e e uv h ' '' ' H '' H ' '' ' H '' H Uig he heoem.3. e hve : ) ( A ( ) A ( )) F( u )( A ( ) A ( )) F( v ) J J q J k j q IH eq F u ' '' F u F v q ' ' F u F v q j '' '' k Cj Ck We emk h '' '' imlie Cj hece. Hece e hve: J q J k j Ck k j k J J k qj j F u F v q F u F v Cj q Ck F u F v q k k k qj k j F u F v Cj q Ck j k F ucj F q vck k d Uig he domied covegece heoem e hve : 85

4 IJIET - Ieiol Joul of Iovive ciece Egieeig & Techolog Vol. Iue 3 Mch 5..ijie.com jk jk J' J'' ( F( u ) F( v ) d q Ck The um j k No J' J J'' q J ' d J ' qj' ' k Cj e fiie. J d he iegd. covege o J q J k k F( u F qj Cj u ) qj q Fv F Cj q Ck qj q Ck v / / / k q q q M / q No e hve : k F F u Ck j F qj k k A A F u A A F v F u F v A A F v d d A A F u F v d Cj We ume h ll hve he me i of A A move i fo uig. The he hough i fo of he i he igh d die. We ume h u d v hve ooie iie h iu u u. The Fu F u d A A F v Ck F(u ) h he i h he i F v. Fo F u A A Fv Ck F u F v o v Hece mu hve he me i. We geelize o uv i iig u u u d v v v. ice k k j uig heoem.3 e c coide uv h d ; ih ; ih Hece: k F k Ck A A F u A A F v F u A A F u d j F A A F u Cj j F v d k I he me me e oi he vlue of: A A F u G A A v ) 3) G A A u G AAv 4) A A Fu Hv 5) G A A Fu Hv 6) Hu Hv 86

5 IJIET - Ieiol Joul of Iovive ciece Egieeig & Techolog Vol. Iue 3 Mch 5..ijie.com We ummie hee fomul i he folloig.we coide hee oiiliie ecl FGH d if i oe of hem e defie : F A A F G G A A H H ih fied. We coide of hem d d e hve: u v F F F u F v k u k k v k u v d * k k k hee: F F G G H H. o kf Ck kg kh k F G H F G kg Ck kh k k F k k k k * * * k k k k k We ecll h N oe em: We coide F F hee d he H e hve ol d F F elive o d imill h h h H H G G d We defie M M A A F G A A H fo d i i fo i The M M hee i e elce ih. We coide M M M he M d e emk h he fomul u v i ill vlid ih he folloig coveio: M k M k M M i d e dd lo he d k em: M M M u M v Fo M e deduce: M u M v F u F v M u v k M k k k u M v d * k k k Whee i he lce of FGH e hve F G H ummig ih o e oi: M u M v M u M v F u F v M u v k M k k k u M v d * k k k 87

6 Hece: IJIET - Ieiol Joul of Iovive ciece Egieeig & Techolog Vol. Iue 3 Mch 5..ijie.com M u M v M u M v F u F v J J k M k k M u v u M v d k k [7]eecu C. Femoi ochic Iegl of Coiuou Pocee/ Alele Uiveiăţii Bucueşi g ; Cii-Miel eecu Degee:. Bchelo Degee i Mhemic Mg Cum ude Uivei of Buche 979. Me Degee i Poiliie d ochic Pocee Uivei of Buche PhD i Poiliie d ochic Pocee Uivei of Buche 997 Plce of emlo:. Uivei Poliehic Buche 4. Cocluio The eed o uild he o-commuive Mkov ocee give he evoluio of oiliie i quum mechic. e uild hee ocee o immeic Fock ce hee e do o hve eoeil commuive veco d hee he commuive oe doe o occu eee oeo deciig dijoi ime ievl[5]. Fo hi eo he ocee e oied oluio of ochic iegl equio. Thi mhemicl model cee he oiili o couc hicl ocee ochic iegl equio oluio eig he me ime e mehod of ovig h cei ocee e ocommuive. The model m e ued i diffuio ocee. Cofeece:. WEA. INAE 3. Euome 4. Keoe Joul:. Hidi: Mhemicl Polem i Egieeig. Bulei de ciece Mhemique Roumie 3. NAUM: Ieiol Joul of Pue Mhemic 4. Alele Uiveiății Bucueși Refeece []Be C. ee R.F. Wilde I.F. The Iô-Cliffod iegl Joul of Fuciol Ali 48(98).7-. []Hudo R.. Phh K.R. Quum Iô' fomul d ochic evoluio Commuicio i Mhemicl Phic 93(984) [3]Aleum D.B. Hudo R.. Femio Iô' fomul d ochic evoluio Commuicio i Mhemicl Phic 96(984) [4]Hudo R.. Phh K.R. ochic dilio of uifoml coiuou comleel oiive emigou Ac Alicd Mhemic (984) [5]Phh K.R. Bh Rjm B.V. Mkov dilio of ocoevive dmicl emigou d quum oud heo Ale de l'iiu Hei Poicé Poilié e iique 3(995) [6]Cuculecu I. Oe A. Nocommuive oiliie Kluve (994) 88