Về các nhóm đối xứng không thời gian mở rộng của lý thuyết trường lượng tử

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Ng\fO'i nhen xet 11

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an se ~

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vao hbi

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bao vi tid gOi d:~ng chlIm 1$
¥ h9c ~Ang h9$>T.P. JIb Chi
nga;y
thq
nibn 1989

-

Co 121' t1m hiAu lu'n
Tang hC1.P~~
'gia.
.
a

,

-

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.8n Nha

J4iI1h

an tCl-i !lhll vien 'lrIICIng d:~ h9c

phS gb Ch! I[~

va t~

']hit vi~n

Q~c


.

fIij fJlu

lincm doi ~
khcug ihc,i glan can ~n nhat Qua 1;
thUY6t tI'lZCmgluvng t& tU'C'ngdot t{nh la nhoa Poincare" .Doi
me; Poincare xu§:t phat t~ quan ni~m v~t ly cho rlng khong
.
, ..
",
..
,
thai gian cua ly thuygt tuong do1 hiP
la dang nhEt va d~
hll6ng, vi vb cae d~c trtmg e~h~

lll9'llg t~ khong dU?c
tbay d~l khi tlnh ti~
bo~c quay h~ t9a dQ ichong then. gia.'1.
..
..""........
lfuo~ P01ncace
co 10 tham so, cae vi t~ Qua no la ceo tOaD

tu. 4 - xung l~g

Pp.va 4-j;enXO'1110
men xung luvng ~;"V

..
Nhom conformal

,.."
Ie Mom

.

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ioincare

d01 Dmg ch~a Mom
nhu mQt Mom con. h"homconformal la nhom co 15 tham 80, cac
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,
vi ill cua no bao gam 10 vi tll cua MOil. poincare V8 ngoa1 ra

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C9J1 co vi tll D wong

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ung vc11. phep bien

-

~

,

d01 co den tO8 d9 T8

KjA tlJ\;1ng ling vai phep blin d~1 conformal
die trllllg.
Nhoc
conformal
Is nhom r~
.nhSt g1lt ~t bi en hinh non anh sang..
G:i.~

thUt

~t

bien conformal

ngb1~m :tEh1 Xf hptolD-had1"Onpbi
ido. r§ng

nhom d01

ntng

c, neng luvng cao d~c
Poincare

cac thlJc

S8,U :mc nh~n.
b~ng

~ch

d\l8

VEtO

cac ph.p bign dAt doi xttng 'Ych cac tham 85 spinO' dan den
Me):; sieu dot x~g. 'Sieu dO't Ding ket h9'(>cae truang
..
..,~"",..
...... boae
Va ferIti vao cung mpt da tuyen dot xung va do do .dan den
MUng khe Dang to ldn d~ xay dlJng ly thuy't tb6ng i:1hSt c.ac:

,
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tllO'ng tee cua cae b,t eO' bBn. cae ly tbuySt traong
.~~iu

d6i ximg co mpt as t:!Dh chat 1; tbUI SO'phin Q g1~11 1'0 rit
so vGi cae l;tbuygt
kbong siiudO't %ling tm:m.g ang, ob
,

t,

..

~t beng 80 ta1
.

vacuum tr1~t

Ji,

.~

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.

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dhuan hoa h~,
tat ca cae gian db vong
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%Un«
t1eu, trung binh Tacuum CU~ tenxa n!ng

luvng tr1ft tieQ v.v

HIe ~Iu cae 1i tbuyft 81~u

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giG gh 4'~Dh'~ DUt't h1,D r~h ~1't1 cgng trbh 11 thl1ye1:
Db'. 11&. hf .1,~o dd1 =g

-v6i tbltC 11~1i. c~ vit 1i h~t

Ilguds. ta olio rh8' s1iu 461 x.ing ta 161 thc1a' - 'th,. chi
h1fn 11&~Ii dU1 Db(t
cbo mic v~ d'a' J18l1gt4J. aliA 11
tbu1it th5ug Dbgt ~wng taer' Steu d~1.g'
si' BP dit
m§t

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trft -1:\1.0 do trons ph~ hOt, bOb chI ad t12ams6 tdl y cqA
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dUVc kIlla oph
thu1'1: 'boD« Db't

thuft' Qua vh cft h1e:tuc07 ~~~. 1i
1cfn (gred unU"1c:atj.on)-, V.V.. .0$0

1inttyet1:12&Dg
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t~

.

t,*o 121fl14.1 ~4t ~" ~iiu hS~

(supergraY1'ty),
sUI1 day Csuperatring)
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thee y t,aoDg 81@a dQ1%Qbg;

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trong 30(2). DUngpb~})thalli '0 120. ttt{~ bcrp s1.~u-~*1Ii.2t
vin 4pg phllmig phap 0$0 MAtt U3n cA. .t1~-~
.clc~ .
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'toe

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tAng quit,

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phb o.

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a~ "'~cblt
Chi nb't cac li thu;ret sleu dol xUng';'
r§.t:g
2 cbUa 481=g
trong SU(2). JIbe, hlnb tbde lufn ~;rtcac

bn>ermul'tiplet

cU: s1eu dO1

.

JJ.

2 chzqe 110 1;. bO1. cac

s1€u tralmg tU1 t:Coh - cie 8ieu truang kboag bt rUg
~Qc
d01
bb1 cae di'u klin hincbe cO t:Cnh ch!t ap dlt. Do do.'
vcn eac sUu tr~g'
gial ticb. co tb' 8P dpg phuong pbap
Jibu steu tm~g. tAng
~
;"" tbu c'c 'hi ihac l1'n..'" hi cic beB!
tllO'ng quan h.l diem cua cae t1'llC111g

thaw pbb. ltet qua, tac
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gt'8 tbu d.
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cae 'bi tbllC g1ils eae he. tvcmg quaa.ba1. cHell
.

cue. cae t~

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thanb p.bln trOl1g .cae h)'perDlIUplet
Sohn1us va Howe-Stell
!oWD8end.

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doi mg
~~c

J'a;ret-

-

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Xhao 8at viic

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,_if

_t

if;

,

hvp dQ1 DDg conformal TO1. sUu

th@h dol :dng rOng h(l1UDot =g

superconforllBl.

~;

neu V$ cllling m:LDh minh d\ sau v\ tiell cbu~
hl~p
bien' 8llperconforml:
Hlu 119.t Lagrangian
slell do! Ding,
v~
dang thC1. le. b1~p 1:d..~ conforl!lal tbl 1; tbllyet; le.ld~i Dng
superconformal.
DUng 1.. v{ 41\1II1Dh b9a, tae gi8 xet
d..nh
bS:t bleD conformal
va superconformal
cua aO hl.nh TO hu6ng
sih
deft :.Ding vcn tv tllO!lg ,tee gs3 .va 88 h1nh dieD dOng
l~c sielld01
a"

xttng cuB 188S. Zumino.
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- De t1111h1el1 cae die hnn cua cae 11 thu1et
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co dol :nmg eonforll81 va 8&q1.r83formal
bt pba Ta

tee 818 .i d' xuSt va kb~o Bat 1110
binh 11 tbll1't

'..
co dol

'
nmg

..
trao-'Ig
tv phtd,

trag

trong siell khanggian
(2,2)
8uperconformal
b+:t
pba va t~ pilat nhc, t~D tf1 nghl~1tt c~ c:l~n lo~i l.nstanton
t;~ pJt~
tthl~ t~g.6.pha r"s va tv phat d61 Ding, 80

:;

.,

hUb. .~ ta cae fhlDs c1Gs 1IIfIt8to"& aO1;~

aOt ~

.

:tera1ont\tcmgtaa ..

miDg. Ph1lCmgtrlm

~t~1Q11
1_0.

cua boeQl1 CAP D8h1faa ch~
~

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trong .~

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11

v~

4~1

inatanton
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xac 9'a tIlc1Dg \fIIg
.

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cua :tera1on.
,
Cae kIt

bo~

D'tiauaft c ~s~u

.-.,

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qua cua tae gia Ita. dktgc c&ng b8 trong

bao tran cae t8P Chi. o"Terpchi T!t .Ji", "Czechoslovak

"

5 ba1
...

Journal of P1Q-jics", "~eng baa ihoa h9C Dp h9C tADg hClP
~8nh ph611& Chi 14inh" (xem b~ danh JII.\1C
cae c6ng trlrm

GOngb8
cua-tae gi; theo 4A t8.1 1$
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ntV).

Ban
1*
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an

0,

eu8i ban tan tit

an, Dgo81phh .& ~u va k't 1utn, CO5
"
chU'ang, J php.1\1C J tg!a °113 trang", Sau ~ 1a n~1
dung tt.

tat !cae Ct.oIc1Dg
cui 1uen an.

+

+
+

6


Cb~g
,

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mfOjt001 D}.riG, POINCJJiE
n'clc
ilf R~rm dA 1;0
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mane die trlln~ uhf!' mon V$O 11nh "vc
ne,b1& c~ trong &) ~~ Sia tr-lnh ~y cae kiln thuc ca b6D
Ch~g

~

v~ t:b&I d8r
Poincare
con:f'o.rmel. %thom81311 d6i

\Fa cae ',1DC1rfmg

clla no I 1110111

Ding vA nh~tn auperoonfor~81.

,~.
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,
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1.- ~~J?o1nce.~:
lbomPoincare t <>ngquat P 1a
'tIt ~ ole Ph~p bign d~1 tw98d~ co dvng:
,

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V

,
Mom

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+.p

(I.1)

'trong do
11 1.119.trfn c~ pbep b1Sn d~l Loren
~,\, 1e Yecta tbh uSn 4- *98 dQ. Ibom Po1Acue

t6ng

qUt

la

htPP~

Jabomcon cui P @1I ~e phep bi en d~1 (1.1) ..61. ~etl\.

va ~~1. 1h,6Ephil Vf1l n8ng
u.~o'

T:i.en ~
#
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bS::t bib

cue MOlD Po1Jtcare

hfP ki h1fu

tllO'ng dot eua lith uygt lU911«
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tLlO'!lg
dQi tum d.

phat b1E!11Mil 8au: ,"V()1. phep 'tie

Po~nca"

treng

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u~ 1;a

htp,
.

v.cta

1

thai

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cue ahOmcJo.

bien cfAl thee bl~u c1dn

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Pi"

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cae

,

Uh t,c
la Dboll 10 them eS,
Vi t~ 1i. 4 vi ta tb;m tUn va 6 vi t&i quay.

,-,

tit
.

clol

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dall 40 -c,agi8 trinh bBy dQi 80 cac vi tll. cue JIhOm
rQincare,

,

cue

,

phUO'J1gPh~p pn6n 1091 ~e bl~u dien
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cae

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tr:1, rieng

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nhq~
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CO hal khong gietn bieu d1e1l khac May 't~g
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v(;i
ha1
bi~u d1'In 1Ih.ang.'t IIO'Agduong cue SI,{ 2. C) , do 40 cO .hat


10fl spinal 8pinO' oh1ral-'t~1
'i~!-;6
khaaS- ~-d;i;)'~
sp1nO'Ohiral phs1 (oh! 84 cO eft..). Cae eptAO' ha1. thUh
."phlD

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do <111110"
891 11 cae spina \Yeylvi (flu Ueu cSuve 'fe,l

3

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duzag de .a h nO't1"1D6.

DW1 pbip nstJJch If~ toe. 4~. spina

m

cbi

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ob«. tren ct1\1,ln thanb ~tDa '9'c7tchI ,88 cO ch& da!\ va ,
n~
1,t."£I~Tiftpbep btln ~~1 00 t{Db e1tft l!-6 c!uO1.

dfDg lI't phep blln. .
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( ~p:1J1O'. )
biE;U dlen Va kbao 8at ceo spIna 4 tb8Dh p~
hong tNCmg bvp tADg quat bt8ptDa cO 4 tbUh phh phtic Ii-PI day la tr1img b9P cae 8p1DO'Dirac. 111\1 b& sung them
cU.'u IclVD tbU'C llajorana
1'c;r C 1f'T 't' 9 tbl Qua' co 4

.

thanh

an

phh

thyc

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"r~g cac clot hot v,t Ii t~ng qqa1;Dhlt_hJD dl' 01\1 trac 0:
- G chi co th4 Ie. t{eb trtIc t11p cua nbO.>f1'otncaft
v~ .
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J.- BbOID819'1 461 ~t
i)j .;. rOag !tho. 441 ~
'oincban" cSlu DgWt1;a *l~ Dbo. G obaa Dbo. Poincare,,;"
UJO1;

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Dbom

t

(spina.

cl61-xtiag

trong.

cac. cI~

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d

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Iuy Dbien cO tl1l d1 D hOD*c c8c11ad ~
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cac %1<1108
G Idt'angpb81 la cac nM. Lt. thm.S,.m:mg - 1&
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cae Mom Lie phan biG (graded).
Mn cOM cac" Y1
~.(thu9c pha. biG 0 '9'4 cI!IIJO&91 14 cae

~" CUB Dboll P01noare

Vi t-i ch~n) dua them cae 1f1 t~ $piDC1(Ct. . Q. Hth*
pban ~o 1 .:.' dJ.J9'C~1 18 cac vi 1;~ Ie). Dt1' as ma rQng,
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g8m oa giao bon tit .. pttan g1ao hoeD tu. lChoa tmmg tJD8 'tCi'1
ct;1 aO nay dll9'C ~1 1.8 nbO:mdeu dB! xUrig;

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vaG
eu ten cua del
80 nay co net d~c Di~t

Sau do tac"l1i

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401 xung.

"1a ca'; boson va cac :fermion n~m trong cims m9t sUu da. tUJ.I~;J
va cO cims 11116:1
luvng.
..
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.
4.- ~ieu khBn~~~ - s~i~ t.rsmpg:
..Lrongt;Leu
teie gie. trlnh 1187hbh tbalC 1.

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s1Cu

tmang dot vc11 slall doi %Ung III1nl11a1 vat 1 vi t{, splnO' (N.1)StEIl kbang sian ls kbqg1an
m& rSiDg cb.m cac t08 d9 khong
thO1. g1an thOng thuDon.s . va c&e toe d Q spinO' ph8n €1ao been.
BUu traang ls cac 'tzuOng ne dpm trang
SUIl ][bong gian.

Hlnb th&ic 1ujn. slau
tuyen

slell

lofD

b(t

.Ugp tbeo

dol'

traOng cho phep mOt~ 89n gbe

%Iing. 1.egrangian

b(t

bLh n cac guy tIc Fe7Dng& ngii 8:!.e~ truoemg.
~.
.~

bien,

.

1i tbQet

cae

eta

nh1'u

cO th~ phat b1~u trio

5.-, Jgom
con1'orMl: .111Jbomconforlla1 cu8 kh6ng thO1 glaD
,~
d01 't98 d9 bao g~m:

..

18 Dhomcac ~ep ~~

- cac phep bi&n-a&l cue Mom Foincare,

-

Phep Mill

-

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Phep bl.Sn etAl eon1'orma1 dIe trung

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dA! co din toa

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D

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dQ ][bong thCn. g1an

-~

x2),

tT(-x}

.

(I.2)

1-=.-c2x2 (I. 1)

lIhom conformal co 15 them 86. cac v1 tu. 141'i1,&\'"
~
,
I \

va K}\.

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Sau do tae gi~ trlnh 18y d,1 86 cao vi tal, gisi
b8y
thIGhvat Ii oao pbep'b1fn et~l (I.2) ya (I.3>,.tr1nh

,
phuO'Ag phap

II
Jt
cae bieu

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dien cam 8inh cas Maclcey, Salam de
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a
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ao dtnb quy lut~ bi.u do! cna toen hr tmcmg doi YC1 cae
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phep billU.l 401 cua ahom confer_l.
II

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6 - ~O. auperooafo~, 1IItOa
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ta pb8t t1tb yao 481 a8

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ooafo1'1l812 n tw
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cae hi tl..m bop YJ _0 .tJtb cf41 84 8UpU'OOllfOr8alII.
ap1DG'S~.
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sIiu WI XIfBGJ! ROBO

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