Canonical Quantizing of Spinor Fields: Anti-Commutation Relations

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Leslie Quentin Fox
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1 JOURNA ON POTONICS AND SPINTRONICS VO.5 NO. MAY 6 ISSN Prn ISSN Onln h:// Cnonl Qnzng of Snor Fl: An-Common Rlon D. Grn PhD Unvr of Brln* Ar Nw mg of hr nor ro on h lr nghorhoo of h -m nor lnk-wv r lh hrogh om n-ommon rlon n ron n nnhlon oror n nonl qnzng. kw on n h orronng nor frm hr mlonn n fnon of h oror. Kwor Cron n nnhlon oror Dr qon lr mg of nor ro nor lnk-wv. Thn hr mlonn ml gv : whh fnll m whr nr o k n onron h n-ommv rl wn h fl n on R I W I. INTRODUCTION E onr h grngn for lf-hn Wl fl ]: m whh h onjg momnm o h lf-hn fl ] * Drh Grn o rofor of h Drmn of Morn Ph n Unvr of Brln. long o h Rrh Gro r Dr.Bln rnl mmr -ml: grn@rln.n.. Rmmr h w n o gv grngn h m of wo grngn omonn on o frmon n ohr nlng h Mjorn o know f I g 5 m B For ml o ornzn olng n h -m w hv B n for h Mjorn nrno ov I B I om gvn : I B ] o homogno n oro Frmn-Roron-Wlkr gomr whr B hr Bnh omonn. or } } } Thn ng I w g or qvlnl For } } omonn Dr fl w fon whh lo n wrn whr m m

2 JOURNA ON POTONICS AND SPINTRONICS VO.5 NO. MAY 6 ISSN Prn ISSN Onln h:// n h orronng nonl n-ommon rlon r } } whr h mr Dr mr of h form Thn w fn m m m m 5 whh lo n rv rl from / In n nlogo w w n o wr h grngn o omonn Dr fl : m m T T C m C 6 whh how h h Dr or Mjorn fl f h Kln-Goron qon n o h Dr qon h ln-wv olon. D o h oml olon of h qon m w n onr olon of h form whr h vor n r omonn onn nor whh r ro of nnhlon n ron oror wh omonn nor. Th r h nghorhoo longng o h lr n h -m. whr n nwl C h mr: - C 7 n h orronng nonl n-ommon rlon r } C } 8 9 B w wn o fn olon o h Dr qon n h mol of Fnmn-Don h o : II. SPINOR TECNOOGY Th -omonn nor o h fl qon: m m In h r frm w n hoo: n lo m m 5 m whr w hv h Fnmn lh : m m 6 W wn olon o h Dr qon n h frmwork w wn o on lnk-wv rl from h Fnmn grm who ml r ro n lr of ron n nnhlon oror h n h rl nron n h gv h rogon nrg n h qnm wv on. } ]. g / ] g whr h rm m r wh onvnn normlzon n h. Th ho orron o gnvor of h n mr 5 : If or - for m - m I I. wh mr

3 JOURNA ON POTONICS AND SPINTRONICS VO.5 NO. MAY 6 ISSN Prn ISSN Onln h:// S z 7 n mlrl 7 : K K whh r S z S z Th ho rl 8 9 In h Goron frm for rr nor w hv: m m I rghforwr o rv l forml for nor wll no n hm; ll w wll n r ro of nor of h form J z J z ] ] whr h oror n r rl wh n + or own - long h z. In l lo om h rr nor 6 : whh o no n on. W fn m m Alo w hv h Goron n: w g: m S ] 5 m m m m n m S ] W n fn nor rrr -momnm lng h mr h orron o h oo: D K 8 In S 5 whr j j K ] n nh / m. Thn w fn: K 9 5 S. 6 I T I K. S 5 whr h h mr } g. Ang h wo qon n ng h nwh rl hm wn nor w hv: 7 K j K j A A A.

4 JOURNA ON POTONICS AND SPINTRONICS VO.5 NO. MAY 6 ISSN Prn ISSN Onln h:// m m m m An morn l on whn Th rr n o monr h whr n r -omonn nor n n r h ron n nnhlon oror. W wn o fn forml of h mlonn hrogh h qnm oror n hr nron lon o wo rl nrng. An lon o ml fl fll on onrng h rm rlv lm of mv rl whn m h o h qon 5-57 n 6-6 wh h onon n h Goron n wh 5 k h form S ] 5 S ] W wll fn vr fl h n m of h form whh wll n ll onrng h m ovr n rmov ll h mmor of h n-qnzon n h rl n n onl on h momnm -vor n gmm mr wh ll n onr. In h r frm onr m w hv: Th w onl m m m m 6 6 m 6 m 6 III. AMITONIAN CONSTRUCTION TROUG CREATION AND ANNIIATION OPERATORS Th gnrl olon o h Dr qon m whh wl n h frmon frm ]. W wn o fn forml for ron n nnhlon oror rl nvolv wh rmn form rlng wh h nrg n h oolog of h lr whr or h nor wv n whr r fn h lnk-wv of h nor wv gvn 6. W wn wv ml of h form: whr whr w hv mll on h lf of 67. For h rmn onjg w g 8 : 69 whr r m nnn. Smlrl o mllng on h lf of 67 w hv: whr h rmn onjg : 7 7 ] 6 Ung h n-ommon rlon o n w hv: 8 ].

JOURNA ON POTONICS AND SPINTRONICS VO.5 NO. MAY 6 ISSN - 857 Prn ISSN - 858 Onln h://www.rrh.org/jornl/j/j.hml 5 7 } } whr n lo h orronng rl of n-ommon: } } Smlrl o h : 7 } } whr mlr rl of n-ommon r.

5 JOURNA ON POTONICS AND SPINTRONICS VO.5 NO. MAY 6 ISSN Prn ISSN Onln h:// 5 7 } } whr n lo h orronng rl of n-ommon: } } Smlrl o h : 7 } } whr mlr rl of n-ommon r. Alo w hv: 7 } } } } } } W onr h mlonn o h wv fnon 6 : 75 m An onrng h: ]76 m w hv 77 whh 78 Thn ng fnll w fn h mlonn wrn hrogh of h n oror: 79 ] V E whr h rm V E 9 o for m of h zro-on nrg ] of lr fl n oo gn. W m h h zro-on nrg nll onn rm n w n non h mlonn onl : 8 ] Th nrrng of h mlonn ron gvn 8 rl h oon of h h -rl n -rl n rmnn w. Fg.. Sll of - n - rl n of h. To orv h n h nr of h r h zro-on nrg whh nnll whn mor - rl nrng wh - rl r. Th mol ol orron o h lrl hrg whr h r rl h lron r nr o h rl hrg n h. Th -rl l rl ol oron. Th hrg Q gv h lron. 9 V n. k E

6 JOURNA ON POTONICS AND SPINTRONICS VO.5 NO. MAY 6 ISSN Prn ISSN Onln h:// IV. RESUTS - n - rl r ngh h vl of h hrg: Q j 8 whr j wh h wv fnon orronn: 8 Thn onrng vr mlr llon for h mlonn w g: Q ] ] onn 8 whh on h nmr of - rl wh h nmr of - rl. For ml n nron wn n lron n oron h lron wll - rl n h oron wll - rl h fgr. V. CONCUSIONS Th n-ommon rl lh hvor of h frmon rl v n - n - rl n h whr hr mlonn n wrn hrogh of h rl. kw whn w mll h nor h rr nor w g for for mr. Thn h n l m ovr ll nor hrogh of hr mmon rl. Th rll m ovr gn- of S z n mr. Th hr hol no mmor of h n n h nwr hol n onl on h momnm n gmm mr. W lo hv n l form of h nor for. W h n gnrlz on or onlon ov. Th rl : m m Th rolm of h hl n fn whn h omonn of h n n h ron of h hr-momnm rr o frmon wh hl +/ whh ll rgh-hn. A frmon wh hl -/ ll lf-hn. Th nor orronng o rgh-hn frmon for - rl n for - rl. Th h rojon oror wll m h ono h lowr wo omonn of h Dr fl orronng o h rgh-hn Wl fl ]. REFERENCES ] A-Wollmnn Klnr n F. Bln Snor Thnolog: h Fl Formlm o h Dl wn Qnm Prl n hr Wv Jornl on Phoon n Snron Vol. n ] Grn D. Wl In n h Snor Frm n hr Pro wh Thnologl Imlon Jornl on Phoon n Snron Vol. no. -. ] M. Srnk No of Qnm Fl Thor Unvr of Clforn Sn Brr h:// ] J.. Pohm J.. Plmrg J Frnk. J. Colng K. Dvll E. J. ngl A. J. n T P. F. "Slow roon gnr of zro-hoon oon of + n nn lr fl" J. Ph. B

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