Các trường hấp dẫn điện từ và mạnh coi như trường chuẩn trong một sự tiếp cận đại số

•^m Muc Lyc MÓ' BXTJ 1.1 B^n^ h9c va ct^ng- li^'c h9c 1.2 ITinh hgc va ct^À so , A ^ « o « • • « « • « • 2 y^ 1.3 '^"^^ l y tT^i '^"^ • • lo Van die n ^ i i e n GUU - • < • Ilìim Tnfc LU|lT ETQTJAv-^ERI^OlT v l T!?OiaOH ofjA c I c TKTjtìTG T y DO o • • o « 2,1' S)g,i so "biquaternion • • • • ã ô ã ã ^ ã ã o 2,2 ^iauatemion va ten:^o' 12 2.3 B i q u a t e m i o n va% spino' • • « • • • • • • • , • I4 2.4 Gac phutiìi^ t r i n l i tx'ùtoH.-^ GO 6:±èn I6 2«5 ^ ' i so octonion 20 2.6 Phubiig: t r i n h son£r octonion cna li^t t y 2.7 Tiubng vecto' t o n ^ quat 25 2.8 Trùbtig nuclon 30 2»S 'PiiìDlag t a o nuclon - n o , « » ineson , , 23 34 TliDW TIG UÈ? DIH v i TUDKG TAO SlfilT TfP , ããã 38 3ằ! HĐ t9a {è9 va hĐ qtiy chieu ã ã 3.2 The cbuan va trubVig chuan 3.3 ^a.c phufeìig t r i n h trubìig chuan cua tnvxmg hap dan 45 chan Idiong • • • 3«4 48 ^-^c phubtig* t r i n h Einstein - l'-Iasn-rell Tmhia TAO vi^vd età ROGLOIT 39 • • • • • • • 5^ 63 4*1 ^"iau fl" bat bién chiral 4.2 Khon^ gian ngòai va khong gian 72 4.3 Truti^ng Yang - T-iills co khoi l-y.ìD'ag • • • • • • 77 4.4 Hau chuan f 81 4.5 i'^au chuan f - g K 64 y4 Kfe LITJK TII • • • • Lifli THAM EHAO , ã ã ã ã ô ã ã ã 66 95 59 ^mmmm^aaar- 1«1 3oii;-;' hoc va (x^ù'i(: ly.'c h9c uv"t V'"',t 1"^," l a su' l i e n a":) '•:hor,'': nnat ;-.:•/.': "'•" , " ' ^^90, >^o^G noi '•: "Lan a^t .,•: ' - ' V •- •,• cua l y thiiyet ::o ^" -'Sy^ chup.c^ t a h i e u cton^.; h9c hoj;,c hinìi h9c l a t ^ p h^*p fi9i,cau hinh kha dx chaT^ nh^-'^i '^'/j'o'c "•^^o••^^: '':1:T7:'^'-: t o a ;- ••- • - •"- ••- - - - ;- ;• :^ - , • '.-;; •' - : y - bao gom cac phu»i:i.:;; tr.\ni'i chiiyex^L C^v.fy va phiic-nt.* tzin^i tri-tiiig ^ji :-:ot \: j.;;,!' :^i tol ^ '.:.:on, • '"i'^.o cb^n l';.ia cac tip^-i hucn^; o'ia.p n':.;.:: '-^Vc'o ve -^ -^;- ].y 'JroiVV *^l^n d^r • "^ , '190 co 'ÌLÌC- trùfeli:; ten:-:o' phan :amg F , ' ; h^n, to'; : h9c bao r^or noi t r tronf; kìionr '^an / inV-Cvski va tg,p h9;p 11191 iX'^'oìi.'.r \ni tr^i coi nhu' cuy ó:^.o CUT cac h.^/r thu' « -3on'^' Iv'c h o c , jviat iil'iaG l a i thi' lv3p lot) cac oir,^ ccao v? ùxivo'a'': tenxo* i' ,j' • • - - -co '.-ao 'oha.' ^ ^ [ •, • nhlSiK fl^y ^%o va i.r}-t*Qg thor nian - * phutU'^- trxnìi chi'.yén d^n;-'; Lorontz va cac phubìi^ t r i n h txiiJo'n:^: >;a:a?'ell ilien n h i e n , n9i du»',' ohu ^•'•e^-; CUP -x;t l y thuye"^' ".• r- ' qri rion • lire hoc ciia, n o , nhi-ng co mot dieu het cu'c fen • lu^i y l a HV: -ohfit t r i e n ctia cac l;^" th-'ì-'^et vg.t 1-^ l^j tQ,p t**:a"'ji'^ tir.i.ti'c liet e' su' thay t^oi c^ia ^.^^•- thay • ' ^ ' • • • • djrig li/c ' ' • • - / , • - - • • • • - - , • , - • • x r{?jniltonl^n, the:^ - :-^ • - \)Var • - - t a "a l y tuV:;o:" u'ng, -•^ - ' a _ ^ A _ * ^ ^ _ * I t'- o a c c!:g,i / u •l'hoo ^ ' - -~ •"-•," A ' s a u ::;• , *» 90-co«« dien va co' hoc tufctig ^oi t i - V- >^ /• r^i A (1 [] holV/ [] • ^ _ '""^••' , '"^ '"• "":-•••'• vi.^c ,ch9n luj^, làia khon-; i-, -.^^.-": l y „ '^ - ' ; :;;- c n c tut''ng t a o ce' han c u a t\^ r j h i e n , n h vj"' 1:' , cli^'c la*^ ci/,t 'V'T ••^••'; ^.C V " , l o "'"? >*: ;'i9n n t iihom chxu.vn diii/ ^1- :;; chon lu)? '.-^^ • t h U c v o nm^c d:o Tìhù h(^T! e*.-ì oac •' ' ' i tu^^ i.'^t c^.c -• ' '-^: - ,[,] au hol ' V A V f T ' * - '^ - ' •^ A A •V ^ r A ' ' l^ ft :.:ot' bien biot'uatemì' %.-'' s A ^ TIO' •1- r r [•.x-rmt^,; ( h^ -, tflnh ^^atemio \ , co o i u u a t e r n i :.: xemanr .;au.c:-iy- t hàn: mjt ::ien phu'c X t h e cac cau h o i t r é n aùy th^a:'h no-'j A , » «^ 1^ no "-: ':a tt^ìD'c ohin:- :ai l e n rà\ ;u oj" 9n tu' .•",::35S!W3s» •-" ^- la: • : khon ^ -ohai - '' '.:• c:''; ; • • ', :;"ta "1^'ori-, -ian v^t l y la not t i n h chat n9i tgà ovp l y rhu^et , raa -lày sinli tu' nhuV;v '?:ia t h i e t ben n;,oai quy dj-nh boi thu*c n i i i ^ m Tj?ai l ^ i nuatex'nion co n9t v j t r i ii'u txen tron:; cac hg so , xi\kt phat tu' c^nh l y Fx^obenxus : a u a t e r n i o n i^- _o ,-,;.' •.•• phu'c no' r9n ; Titiat cung iP-ang cac t i n h chat nhu' so th^i'c, t r u ' t i n h chat "iao hoan cxia phep ^ilian :.,eu cac tT§.: l\*'n.'"'' vg.t Iv co' bau cTU-ir/ prtax uuan theo "-a^ '^1"':^ ~ac ti-ih t ' nhu' so thy'c ir-^ai t r - ' -.ot r,o nuy t a o , 'cihi A* A" > ' "ohai ohon 30 ^hu'c , fto quate:iniion , so octoriion v y , , 7x t5.nb •'chon.'^; '^iao hoan cua 'S^ì.\^V iihan _& d'info chung n^'hi9in 'ìa,;' '-;u ti-o:iig Ixìih "v'i^'-'c v i rno, non n^u •:,• • \ hep ho cuy t a o viao hoan, thx chì-a" t r phai ch9n quaternion -'"i l y tVlai trer ':3n m^t nli^xi :c:^t : ':^^ phu h^^ cua d; ' ^^ ui'iiiatemxou vo*^, c:?.c hxfu t'ytb'ii ; ax^xi tu' xx>i\z ximi cha'o UQX tg.i hv.-a Ix^u 'liti t n e ciị cua cac ca^u tl•ì^c ag-x so QUI CO t i n h u^'au ^ìhien tre.:' ; l i n h vyi'c ^i^n tu' hay co t h : e' r9n;; sang cac Ixr."^^ - U o r a Ic'x nx9'i cou pay xnu^c • 11 anh iiol ctaag phat t r i ^ n l a v^t l y t^i !•,' ' ^ t l y ag-x co /^.U X op cua r-9t chu7/-Cii ,:o • < ' • ' - ' r 1- fc.],' ' • * • - ' • - * []- ^ n * • • [] [] '.-PO* tb^T'b 'ot 1"" -^.oTi^et co hi'^T' 'lu'c iquatem:.' * ^ , ^:à (ì^^'c ru' l ; ; - 49 A en ora he-V2t 1'^" co ve so bf?' •" ?, coi la, m^t txTAjii,;-; fco , , '.uyet d i f n tu' '* - J A ^ CI ' : '•!• '•il"' * -*• " C'.PJÌ' u• ••• - •'^••ert thoji^:- t:n:t;ìi r - t e t e i o n i c l,- j • •" " vat ^ • ["] ^ •^ *> • A » * A , t ' A A nar^^: vc'i txnl^ V ' /• % ,T"0V";7Ì - * «f V A ' * A V -'^"ù 'UY ' ' D^r'O^'TOS » X t ^ t u cu: /'onioi' ^l^J - C'L A v'- ••• e r:on.: l u ^ t r a n co w: l?-r t ] ^ vx'^c it •-• •• ^ A '' • •_• • / ^ A' A ac -.•ax • " •': Van Ce n.'^nien cuu ^ ru.t ^, r a m9t so k e t lu,; -v; ; ] ] - ' ' • t i e u t r x n h ba^/- t r n , co' tr.e ':•-il '^.' .•t'" f p •' •!9'^ .X' '-.'Cs'ù l''0 ii-'ì'P /S •;ìhubv*^v l i n dong l i / c ì).9c, Tnov " *; ".ã/ oã 1; ã190 cua- l y t]uiyđt truSohg c h u a n l a l y t h u y o t dtiy nliat h i | n c o I d i ' j^',:;.•, t.;G t r c a c wV:Vi*r.-;' t a : 2on-_;' h9c tut'-' n ci":a t-'i' •••hxau t h e o ::.ot r u ^ n •'^-'r^'ì W _ % _A l i nl.iif^rt van l a 0:9:1 ]';9c cxia àior., _• ,an the' • Dio"^ "''hac ' ^ i ^ l a tao' t^.i r e i '"iof-i cu:- 'la tg.p Ic^.oi ; * thox "•.•" *- % •• ^ v'^ ^y^-c .:-' ,3,1 co l u j n an n?.y l a ni9t co F^mr da.u t i e n t h e o hutotig d o Ghi'iv: t o i ^JG chnn-^ t o rèùxT neu tho' l a li'^ot cau t r u c 'l^l :;ó bioua.ternion t h x e."e phutn/r t r x r J i ti-uVjti^; causai tutn^^- un-»; se l cac DhuiD'n.-': t r x n h " - tìinstein d^.i so •• ViaDC:iell ]cet h^p '.'"iic mo' r ^ n g t h o ' t}.\ành iii9t cai.\ txnic p c t o n i o n s e cho p h e p ^lìx V 03,1 chubng l a G>.UV-/JI 'iicp v a o c a c t u t i ^ g t a o ridili • • d^Jt vaai de n j h i e n ciVa , l i i ^ n en "^orri b a ' chuìDti'g chinfi ( c a c chutn?"-' , ; : Ghub'ng toin t.^t c a c k i e n tbj;'c co' b a n v e (kp r.o b i c u c t e r t i i o n V'-ì oct^-'D.ion, t r e n o c n': xn^r l'ir'rr 'i^n.9- a'^à so tnib'a ; ti^i' c a i Ci::"; • trxnl J x r a c ,quen t h u e coìi ouy x^a in^' c u a e r o -nv^irrn-''" t r x n h l a x v ' s l l , F r o c a $ 'ey! Véi io:xub ronc; o c t o n i o n mc-l ma o.':n;!n;"-tox o-"^.c r/ji-^ -:e aon," ^'ai t r o o-'s:ia t r n ^ t'oor.;- , ^Jac :.:^nn c r i ;'^:c.L,M" •loj-,;; n h u vax "cxo c u a p-iutn , t r i n a l ' i r a c 1:1':; •"ac ax^n U.L w'.ì a loy'c'pvj^: bj.nh thi-'C l i i ^ n h i q v a t e r n i o n v a o c t o n i o n -:;"•*•, ^-r.' t:*"' ' d a n v a ui^ìh ••""• • • "Ì: •." ; ' ) txieo ';i7.an dient l y Mihuyet ' ;••• t ^ : c aX'-cbuan i j p c a c phiibUr t r x j i h E i n s t e i n - f-iaxwell k e t ho'pj -'." '"•"•••' Ti' • • ; 'Cu e tho' co th.o' l a d ^ i 20 bic|>i£:,tarùio ooton-i'on v a d a n 'f^en ia9t so h e ' nhi.i' v x ^ c tiv f'ici; ria'! f - ' " "• dàn !n5,iih dS t ^ i t i n b ãã ' ô • , • • - " - -'•' "• thic'' troi ::: d ị - c a c t h e len sx : '•".tvou v-t'.n Thorv : c •a:ing d^rli S7^' U(l,"i) l nhom quay ' Lorentz cùa he quy chieu t r o n g rhung vàn bao toan tenxo' mètric f^^ cÙ£^„ ]±Long l a n t r o n g • Teirco' nang xung lybng cùa trùbTig l'cng-Kills dong v a i t r o n-guon cua t r ù b n g hap lan nt^nh • Chting t a co t h e goi cóc jviubng t r ì n h r4-7-') , (4-77) l cac phubn,-; t r i n l i va (1-70) cv-r, tx^bng nuclon - Yang l l l l s - i';-^ -i-^-' m.n.nh 'let h'-n s y tubng t y cùai h | rhubng trình vói h^ phubng trxnh (3-^'0) mo t cac trù.bnf? Dirac - l'arcv/ell - E i n s t e i n \-', k e t h^v^ thyb dóng chù y 4*5 l'au chuan f - g Viec dim t n i b n g hap dan vào t r o n g mà,u f nliu' thùbng I : t h a y teixxro' m è t r i c phang ngoai b'^1 teuxc-' g nè-i t r e n di-b'c th^^b hi* "^y c-a:^ kbèng gi?ji ^'•à cgn - -f-^ - vào r t dg l a g r a n g i a n (4-^1" so hg.ng 3^jm mo t i!^t d9 La.grangiaii cùa trùbng hap dan PVieo cach Ì,am , giuh cac dj cong cua khong r l a n ngòai va lliong yian tx^o:?'-^ ce ^y' doc \r^rr) tu.'rn'- d i i v c l ^'^irra '•'foi r i è n g r-ot kjìong gxan na:c 00 tho coxig, t r o n g ^aiX l-mong gxan Kxa van phaa^g Tuy nhien , m.ot txnli huong iihu' vQ,y xem r a Ichong thoa daiig bo'i l e r.eu ti'ub'ng hap -ian m^nh chi]ih la sy' tiia>'" doi hxnh hoc Yhonri: gian t r o n g t h i voi tu' ca.ch mot txùbng v|,t ciiat no cu g t a o d^n"^ to'i hìnli ^ * t % A ^ ri ri % « t f hoc cua khong gian ngoai • Co t h e già t h i e t a p r i o r i rang moi qiian he /ciùb, h a i khong gia-n Ichàng k h i t den i-iub neu n9t t r o n g h a i khong /?'ian nià con^^ ^ -'•r.-.^-^ - ^ -ì-Vn yvion - -i'^-n ^ ka ctu'i"'; r-An.v r "i';."'r>''-' • fi^-^r., ITot k i e u quan h^ nJiU v^y giS? Imi khon^ gian da dyb'c Salam va Strathripe Ishar-f , xay li;ng t r o n g mot mo hinh mo^ r^ng l y thuyet t r ù b n g hàp '-Jan goi l l y t ^ i - - t : .-n h a i tercco^ ( hay l Iv thu.yèt f-g hàp 5an ) [x^.^ Xuat ph_at diem cùa chung t a h dSy lliac voi cac t a c già t r e n , tu nhièn co t h e hoan tò.-n dub vào hinh t h u b ' l u ^ n (€ de suàt t r o n g U\i^ de' xay dyn.g m9t l y thuyet trvbn , alman t r o n g 'io •;o hi^.ng kb.ịi lybng cùii f - ineson sinh tu' Lagrangian tubng t o c giùk cac tr^ibng m è t r i c c-ùa kb.Sn;' - i a n t r o n g ^.^à khSng gian ngoai Chùng t a se goi màu l mxi chuàn f-g dÓ phàn h i ^ t vói r^àu chi.vàn f cùa r.yc t r o n g dị 30 h^ng IdiĨi lybng Glia vu tiy A | ^ ^ • f-mịson nSy sinJi tu' chinh sS h ^ g r,c hhU c£ t r ì n h bay -' n^yo 4.2 , chimg t a quan ni|cn vTi t:,:y c^!ia nuclon gom h a i phàn : Idiong ; l a n t r o n g va kliong ^^ian ugịai ^ìnli hoc cùa khàn^r ràan t r o n g trùbng hàp dàn m^uli quy d^nh va dyb'c LIO t a b e l t r ù b n g cac minquat metric ^ l i è n h | vói trùbng- tenxo' f f^ • ? - g.>^ (4-96) r> Hinh hoc bat bien doi vcl i^hep bien doi chuan dinh :c:: : ^ ^ Coi -f-.-^^ry;-,;,.• :^ó va f (4-107) ?v^"=r^.s^; ^ g^v* ^^P^^ ""''^^ ^i^"^' ^^~^^ ^^'^^ '^^•'''^ '^^^'''^ (^-56) (4-SS) , A^ ^-^ " ^^^C 3Ò y^.y t a co : ^ = ^, ^ ^ , ^ ^ l i f ^ , '-^,^$f^ La-^ b i è n uhàn Lagrangian t r e n l n lybjr theo ^ ri, fi (4-108) , B^ •?- t.- dub'c cac -ohubn-: t r i r l i : (4-109) O ( pcj'^^ liB- ^ J $^ (4-llC; co ^ i L ^ ^ r ' - ' % '^(§ ) ^4-111) ^'^^ co-ii.i^^co.rtr] t r o n g : Con -^(s) va "^-^(f) l cac v e c t c b i n u a t e r n i o n thu dub'c Idii l n lyoi; l y bièn phàai cùa o t ^ ilu^.'vyy T (fi theo tf" ^ va di ^TM t^^-.-; ^ cl^Jbb txrii theo cong thub ( - I ) , : + «7^ C*^ ' / ' ^ ^ !.ai inau khac nliau l a mau chuan f 4« f y-, nau abiuan f-g , chu y th^n l màu ;bb t y djng suy r a ~^" hrtnr: tT-inli song octonion t h i co t h e coi l a , bang s y t i e p c3.n dg.! so va t h e o quan diem t r ù b n g cliuan , chung t a da duU r a myt mo hinh l y t h u y e t ve tubng t a c m^nh cua cac nuclon thong qiia sy t r a o dòi '^a'-* "t-i^t g i vo h^ibng -^ , c.ac meson vecto' é va me v^n f S" Ve co' ban , cac k e t qua t r e n , aeu xet myt cach r i e n g bigt,Ichong t h e c o i mei • Trùbng liap da,n coi nhu.' triibng chuan d^ dyb'c 'cha nhieu t a o C'ià xet tf^n Oac phui~ny t r i n i i l'ur^and , c^c phubng t r i u l i hàp 'làn m.^li , mau c h i r a l cZ d^itn ^'ia di^iii 'J:ub vao de g i a i quyet nhung vàn> de khàc cùa v^t l y l y t h u y e t (?' day c a i mól c h i l s y ' t y dong siiy r a cac k e t qua t r e n m9t cach nliàt quan t^'*^' yèu càu tuyen t i n h hoa phubn,^ t r ì n h K l e i n - Gordon t r o n g khung d j l so octonio]i i f c by' UOTP' suy r-, oa- ^•-.et qua , phan n l o ' co t h e so sànli v ó i v i ^ c tu' d9ng f^uy r a mgt so k e t qua tubng t y c.n tu' phubng t r ì n h M r a c Vi và.y co ce SO' (ie nrpl ran^; nSu phubng t r i n l i Dirac d£ t u r a r a t thxch h9:p t r o n g v i | o mo t tubn^• titc d i l n tu' cùa cóc e l e c t r o n t r o n g p h ^ v i r\rniy%n t u , t h i ^hubng t r ì n l i con- octonion co ^-••.' ujjig •^•.- b^^p v ó i v i | c — •' * " -• '- ••'-• ri>^uh 0Ùa càc nuclon Jrrong ph^G) v i h y t nhan , Di ìihien de dieu pliat b i e u t r e n day thy'c sy' co g i t r j càn tim cach g i a i cac phubng t r i n l i t r ù b n g dS t h u dyb'c va day t i e p t i n h tòan t e i pha.m v-^' n^-.r, oac h i ^ u iing cy t h è Trong cong t r ì n h uày , za^àt r h a t tu' phubng t n n h song octonion , chung t a dS l n lub*fc thu dUb'c I phubng t n n h (2-60) cua txaibng nuclon t y , l a g r a n g i a n (4-22) ciia p-aU ie -^an l i e n vo'i cau t r u c xoan cua ichong - t h o l gian tQ,i uhung ia.en cyb nho quy dynh bòi hàp dàn riynh • TTeu g i t h i e t t r è n day l dang t h i càn dyt vàn ite xàv dyng t i e p l y t h u y e t cua chuji,g t a t r è n mot da tg.p ichong - tliòl gian v c l hinli h9G Riemann « C i r t a u fiC TAI Ll|;tl TEAM KHIO G.lT.Yang and R L r i l l s , Phys.Rev 96 ,191(1954) R.Utiyama ,Phys.Rev 101 ,1597(1965) i^.Vreinberg, P h y s R e v L e t t l ? 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':a co h^ thu'c :au: iiay n'i'.; co t h e b ': ' -'•• ion co uio'c e r a Idioni; fl a, - A n Trong t:?5x-;'Dg b j ^ dao M" : ' ^ J^ ; -o ton t^à-biquaterrdon n-ih^ch cua bira"'.atr:)miort K