HAMILTON-JACOBI TREATMENT OF LAGRANGIAN WITH FERMIONIC AND SCALAR FIELD

Transcription

1 AMION-JACOBI REAMEN OF AGRANGIAN WI FERMIONIC AND SCAAR FIED W. I. ESRAIM 1, N. I. FARAA Dertment of Physcs, Islmc Unversty of Gz, P.O. Box 18, Gz, Plestne 1 wbrhm 7@hotml.com nfrht@ugz.edu.s Receved November, 6 he mlton-jcob formlsm s led to sngulr grngn contnng vrbles whch re elements of fermonc nd sclr feld. he equtons of moton re obtned s totl dfferentl equtons n mny vrbles. he ntegrblty condtons re exmned. Pth ntegrl quntzton bsed on mlton-jcob roch s obtned for the system. 1. INRODUCION In ths work we ntend to study sngulr systems wth grngns contnng elements of fermonc nd sclr feld from the ont of vew of the mlton-jcob formlsm develoed by Güler [1, ]. he study of such systems through Drc s generlzed mltonn formlsm hs lredy been extensvely develoed n lterture [3 5] nd wll be used for comrtve uroses. Deste the success of Drc s roch n studyng sngulr systems, whch s demonstrted by the wde number of hyscl systems to whch ths formlsm hs been led, t s lwys nstructve to study sngulr systems through other formlsms, snce dfferent rocedures wll rovde dfferent vews for the sme roblems, even for nonsngulr systems. he mlton-jcob formlsm tht we study n ths work, s led to few of hyscl exmles [6-9]. But better understndng of ths roch utlty n the studyng sngulr systems s stll lckng, nd such understndng cn only be cheved through ts lctons to other nterestng hyscl systems. Our m n ths work s to ly the mlton-jcob roch for sngulr systems to the cse of grngn contnng fermonc nd sclr feld, nd to comre the results to those obtned through Drc s method. Rom. Journ. Phys., Vol. 53, Nos. 3 4, P , Buchrest, 8

2 438 W. I. Eshrm, N. I. Frht. AMION-JACOBI APPROAC In ths secton, we shll brefly revew the mlton-jcob formulton of constrned systems [1, ]. he strtng ont of ths method s to consder the grngn ( q q ) = 1,,, n, wth the ess mtrx q ( q ) Aj j 1 n q q j of rnk (n r), r < n. hen the r moment re deendent. he generlzed moment P corresondng to the generlzed coordntes q re defned s where 1 nr q nr 1 n q he sngulrty of the system enbles us to solve eq. () for q s q q ( q q ) b (4) Substtutng eq. (4), nto eq. (3), we obtn the constrnts s (1) () (3) ( q ) (5) q q In ths formulton the usul mltonn s defned s (6) q (7) ke functons, the functon s not n exlct functon of the veloctes q. herefore, the mlton-jcob functon S( q ) should stsfy the followng set of mlton-jcob rtl dfferentl equtons (JPDE) smultneously for n extremum of the functon: where nd S S t q P P q t n r 1, n 1 n r, (8) (9)

3 3 mlton-jcob tretment of grngn 439 he cnoncl equtons of moton re gven s totl dfferentl equtons n vrbles t, dq dt 1 n n r 1 n d dt 1 n r q d dt n r 1 n q (1) (11) (1) dz dt (13) where Z S( tq ) (14) beng the cton. hus, the nlyss of constrned system s reduced to solve equtons (1 1) wth constrnts ( t q P) n r 1 n (15) Snce the equtons bove re totl dfferentl equtons, ntegrblty condtons should be checked. hese equtons of moton re ntegrble f nd only f the vrtons of vnsh dentclly, tht s d (16) If they do not vnsh dentclly, then we consder them s new constrnts. hs rocedure s reeted untl comlete system s obtned. In ths er we wll study system of grngn contnng elements of fermonc nd sclr feld. 3. DIRAC S MEOD Consder grngn contnng elements of fermonc nd sclr feld gven by ( x)( m) ( x) 1 ( x) ( ) 1 x M ( x) 13 (17) We re dotng the Mnkowsk metrc dg( 111 1). he grngn (17) s sngulr, snce the rnk of the ess mtrx (1) s one. he generlzed moment () nd (3) cn be wrtten s

4 44 W. I. Eshrm, N. I. Frht 4 () Where we must cll ttenton to the necessty of beng creful wth the snor ndexes. Consderng, s usul s column vector nd s row vector mles tht wll be row vector whle wll be column vector. he usul mltonn s gven s or nd (18) (19) (1) 1 1 ( ) ( m M ) 13 () Eqs. (19) nd () led to the rmry constrnts (3) (4) resectvely. hese constrnts led to the totl mltonn or (5) 1 ( m) 1 ( ) ( M ) (6) Accordng to Drc s method, the tme dervtve of the rmry constrnts should be zero, tht s { } ( ) (7) m { } ( m) (8) Eqs. (7) nd (8) fx the multlers nd, resectvely s ( m) (9) ( m) (3)

5 5 mlton-jcob tretment of grngn 441 Multlyng eq. (9) from the rght nd eq. (3) from the left by, we obtn m (31) m () ( ) (3) here re no secondry constrnts. kng sutble lner combntons of constrnts, one hs to fnd ll numbers of second-clss ones, there re nd 1 (33) (34) he totl mltonn s vnshng wekly. It cn comletely be wrtten n terms of second-clss constrnts s 1 ( ) 1 ( m M ) 1 (35) he equtons of moton re red s { } (36) { } (37) { } (38) { } M (39) m { } ( ) (4) get (41) { } ( ) m Dfferenttng eq. (36) wth resect to tme, nd usng eq. (39), we hve M (4) Substtutng from eqs. (31) nd (3) nto eqs. (38) nd (39) resectvely, we ( m) (43) ( m) (44) Substtutng from eq. (3) nto eq. (41), one obtns (45)

6 44 W. I. Eshrm, N. I. Frht 6 In the followng secton the sme system wll be dscussed usng mlton- Jcob roch. 4. AMION-JACOBI MEOD he set of mlton-jcob rtl dfferentl equton (JPDE) (8) red s 1 1 ( ) ( m M ) (46) (47) (48) herefore, the totl dfferentl equtons for the chrcterstc (1), (11) nd (1) re: dd (49) d M d (5) d ( m) d (51) d ( m) d d (5) he ntegrblty condtons ( d ) mly tht the vrton of the constrnts nd should be dentclly zero, tht s (53) d d d d d (54) Substtutng from eqs. (51) nd (5) nto eqs. (53) nd (54), resectvely we get the followng equtons of moton: (55) ( m) (56) ( m) (57) From eqs. (5 5), we get the equtons M (58) ( m) (59)

7 7 mlton-jcob tretment of grngn 443 (6) Dfferentte eq. (55) wth resect to tme nd usng eq. (58), we obtn M (61) 5. CONCUSION In ths er we hve nvestgted constrned system of grngn contnng fermonc nd sclr feld usng Drc s mltonn formulsm nd mlton-jcob formulsm. In Drc method the totl mltonn comosed by ddng the constrnts multled by grnge multlers to the cnoncl mltonn. In order to drve the equtons of moton, one needs to redefne these unknown multlers n n rbtrry wy. owever, n the mlton-jcob formulsm, there s no need to ntroduce grnge multlers to the cnoncl mltonn. Both the consstency condtons nd the ntegrblty condtons led to the sme constrnts. In mlton-jcob formulton, the equtons of moton re obtned drectly by usng JPDES s totl dfferentl equtons. Also, t s not necessry to dstngush between frst nd second clss constrnts. REFERENCES 1. Y. Güler, Il Nuovo Cmento B17, 199, Y. Güler, Il Nuovo Cmento B17, 199, D. M. Gtmn, I. V. yutn, Quntzton of Felds wth constrnts, Srngs Verlg, Berln (199). 4. K. Sundermeyer, ecture Notes n hyscs 169-constrned Dynmcs, Srnger-Verlge, A. nson,. Regge, C. etelbom, Constrned mltonn System, Accdem Nzonle de nce, Rom, Y. Güler, Il Nuovo Cmento B19, 1994, Y. Güler, Il Nuovo Cmento B111, 1996, B. M. Pmentel, R. G. exer, Il Nuovo Cmento B111, 1996, D. Blenu, Y. Güler, J. Phys. A: Mth. Gen. 34,, 73.