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Gauge covariant local formulation of bosonic strings
Nuc1ear Phy51c5 8268 (1986) 125-150 • N0rth-H011and Pu6115h1n9 C0mpany
6 A U 6 E C0VAR1AN7 L 0 C A L F 0 R M U L A 7 1 0 N 0 F 8050N1C 57R1N65 A. NEVEU and P.C. WE57* CERN,
6eneva,
5w1t2er1and
Rece1ved 1 Ju1y 1985
A free 6050n1c 5tr1n9 f1e1d act10n 15 c0n5tructed that 15 10ca1 and 1nvar1ant under the 1nf1n1te 5et 0f 9au9e tran5f0rmat10n5 c0rre5p0nd1n9 t0 the c0nf0rma1 9r0up 0f the tw0-d1men510na1 w0r1d 5heet. 7h15 a110w5 f0r a 5tra19htf0rward 9au9e c0var1ant meth0d 0f 5ec0nd-4uant121n9 5tr1n9 f1e1d the0ry. 7he meth0d 0f exten510n t0 the 1nteract1n9 ca5e 15 pre5ented.
1. 1ntr0duet10n 7 h e c1a551ca1 6050n1c 5tr1n9 15 de5cr16ed 6y the act10n [1] A
1 2~ra~ f d 0 d * [ d e t
0~x"0~x~] 1/2,
wh1ch 15 1nvar1ant under tw0-d1men510na1 reparametr12at10n 0f the w0r1d 5heet. F0r the 0pen 5tr1n9, x~(a, , ) 15 def1ned f0r 0 ~< a ~<~r w1th 60undary c0nd1t10n5 00x ~ = 0 at a = 0, ~r. 1t 15 c0nven1ent t0 mathemat1ca11y extend th15 ran9e fr0m - ,r < a ~<~r 6 y 1ett1n9 x ~ ( - a, ~-) = x~(a, ,). 7he c105ed 5tr1n9 15 def1ned 6y x~(0, ,), -~r ~< a < ~r, per10d1c 1n a w1th per10d 2~r. A5 a re5u1t 0f th15 reparametr12at10n 1nvar1ance 1t 15 a c0n5tra1ned 5y5tem. 1n the c1a551ca1 ham11t0n1an f0rmu1at10n 0f the 0pen 5tr1n9, the c0n5tra1nt5 may 6e wr1tten 1n the f0rm
L.=~ra~
f,,
(
1 0x~)2,
d ° e - 1 " ° P~-~ 21ra~ 00
(1.1)
where
8A • ~(a) •
p ~, = -
-
0x~ ar
k ~, = •
7 h e L.•5 06ey the a19e6ra
[Ln, Lm]=(n--171)Ln+m, * Permanent addre55:K1n9•5 C011e9e, L0nd0n WC2, UK. 125
(1.2)
126
A. Neveu, P.C. We5t / 8050n1c 5tr1n95
wh1ch 15 ju5t that 0f the 1nf1n1te-d1men510na1 c0nf0rma1 9r0up 0f the tw0-d1men510na1 w0r1d 5heet. 70 f1r5t 4uant12e the 5y5tem we ad0pt the u5ua1 c0rre5p0ndence 6etween P01550n 6racket5 and c0mmutat10n re1at10n5 and take [x~(0),x~(a~)] = 0,
[P~(0), P~(0~)] = 0,
[x"(0), P~(0~)] = 1h ,/~t~(0 - 0•).
(1.3)
7he re1at10n5 are repre5ented 6y the 0perat0r5
x"(0) = x " ( 0 ) ,
6 P"(0) = -1h3x~(0)
.
(1.4)
7he 5tate 0f the 5tr1n9 15 de5cr16ed at a 91ven ~- 6y a funct10na1 0f x"(0), +[x"(0)]. A1th0u9h the ham11t0n1an van15he5, 0ne may u5e the 4uant12at10n meth0d5 0f c0n5tra1nt5 d15cu55ed 6y D1rac [2]. 1n th15 paper, we 5ha11 91ve a meth0d 0f 5ec0nd 4uant121n9 the 5tr1n9 that 15 free fr0m c0n5tra1nt5 and p055e55e5 9au9e 1nvar1ance5 c0rre5p0nd1n9 t0 the 1nf1n1te d1men510na1 c0nf0rma1 9r0up. F0r the 0pen 5tr1n9, 0ne 0f the 1nf1n1te 5et 0f 9au9e tran5f0rmat10n5 w1116e Yan9-M1115 tran5f0rmat10n5, wh11e f0r the c105ed 5tr1n9 they w111c0nta1n 9enera1 c00rd1nate tran5f0rmat10n5. 1n th15 way, the 9au9e 1nvar1ance 0f the 5tr1n9 c0nta1n5 a11 prev10u51y kn0wn 9au9e 5ymmetr1e5, and 1n the 9au9e c0var1ant act10n, 0ne f1nd5, at the 11near12ed 1eve1, the Yan9-M1115 and E1n5te1n act10n5. Up unt11 recent1y, the 5ec0nd 4uant12at10n 0f 5tr1n95 ha5 e1ther 6een perf0rmed w1th c0n5tra1nt5 6e1n9 pre5ent 0r carr1ed 0ut 1n a 91ven 9au9e 5uch a5 the 119ht-c0ne 9au9e [3], where the c0n5tra1nt5 have 1n effect 6een 501ved. Quant12at10n u51n9 8R5 techn14ue5 0f the 11near12ed the0ry 1n a 91ven 9au9e ha5 6een d15cu55ed 1n ref. [4]. 1t 15 p055161e that 0ne can, 1n pr1nc1p1e, d15c0ver a11 the pr0pert1e5 0f a the0ry 6y 4uant121n9 1n a 91ven 9au9e. H0wever, w1th0ut the a6111ty t0 u5e a11 the w15d0m ac4u1red w1th 5ec0nd 4uant12at10n, th15 may 6e d1ff1cu1t.An examp1e 15 the c0mputat10n 0f an0ma11e5 wh1ch are a65ent 1n the 119ht-c0ne 9au9e; h0wever, a carefu1 5earch 0f L0rent2 1nvar1ance w0u1d 5h0w that 1t 15 v101ated when 9au9e an0ma11e5 are pre5ent. A150, the wh01e 5u6ject 0f n0n-pertur6at1ve 5em1-c1a551ca1 phen0mena and 5p0ntane0u5 5ymmetry 6reak1n9 ha5 up t0 n0w 0n1y 6een deve10ped 1n the 9au9e 1nvar1ant framew0rk. 1t 15 a150 p055161e that the f1n1tene55 pr0pert1e5 0f 5tr1n95 may 6ec0me part1cu1ar1y apparent 1n a c0var1ant f0rmu1at10n a5 they d1d 1n the ca5e 0f 5uper5ymmetr1c the0r1e5. 70 5ec0nd 4uant12e the 5y5tem we 5uppre55 a5 u5ua1 the • dependence 1n the wave funct10na1 ~p[x~(0)] and re4u1re an act10n 5 wh1ch 15 u5ed t0 we19h the Feynman path 1nte9ra1. 7he vacuum-t0-vacuum amp11tude 15 91ven 9enera11y 6y
f
exph 5,
(1.5)
A. Neveu, P.C. We5t / 8050n1c 5tr1n95
127
where the act10n 15 0f the 9ener1c f0rm
5 = f ~ x , ( 0 ) f(4J [ x , ( 0 ) ] ) + 9au9e f1x1n9 + 9h05t + 50urce term5.
(1.6)
1n a c0var1ant f0rmu1at10n, 5 w1116e 1nvar1ant under 9au9e 1nvar1ance5 c0rre5p0nd1n9 t0 the tw0-d1men510na1 c0nf0rma1 9r0up. We w111pre5ent 5uch an act10n f0r the 11near12ed the0ry. F0r a 11near12ed the0ry, the 9au9e 1nvar1ance5 are a6e11an. We pre5ent the 5trate9y f0r 6u11d1n9 5tr1n9 1nteract10n5 5tart1n9 fr0m the 11near12ed the0ry. 7h15 15 perf0rmed 6y a type 0f N0ether techn14ue, and 0ne f1nd5 that the A6e11an and r191d 1nvar1ance5 0f the r191d the0ry 6ec0me kn1tted t09ether. 7he kn1tt1n9 1nv01ve5 the three-re99e0n c0up11n9. 7he dependence 0f x ~ 0n 0 1n the 0pen 5tr1n9 can 6e wr1tten 1n the f0rm
x~(0)=
~, n=
e1~°x,, "
(1.7)
--00
where x~(0) 15 def1ned f0r -~r < 0 < ~r and x~ ~ = x~. 7he tw0-d1men510na1 c0nf0rma1 9r0up are 9enerated 6y the V1ra50r0 0perat0r5 L, wh1ch are def1ned 6y ,~,
f
L, = 1~ra~
d 0 e - ~ " ° 6 " ( 0 ) 6 ~ ( 0 ) ~/,,,
(1.8)
where /6~(°)=-1h8x~(0)
1 0x ~ F2~ra~ 00
(1.9)
1t 15 we11 kn0wn [5] that the 0n-5he11 5tate5 0f the D = 26 6050n1c 0pen 5tr1n9 are 91ven 6y (L0 - 1)~k = 0, Ln~6 = 0
(1.10) f0r n > 0.
(1.11)
1t ha5 6een 5h0wn [6] that th15 5y5tem p055e55e5 0ne tachy0n at 1t5 10we5t ma55 1eve1, a11 0ther 5tate5 hav1n9 2er0 0r p051t1ve ma55e5 and that a11 5tate5 are 9h05t free, 1.e., have pr0pa9at0r5 w1th p051t1ve re51due5. A c0rre5p0nd1n9 c0var1ant pr0ject0r wa5 f1r5t wr1tten d0wn 1n ref. [7]. 70 ana1y2e e45. (1.10) and (1.11), 1t 15 m0re c0nven1ent t0 1ntr0duce creat10n and de5truct10n 0perat0r5 def1ned 6y the e4uat10n 1
+00
/%(0) = ~.(2a,)1/2 ,,=E~ 0 / n, e 1,,,.
(1.12)
128
A. Neveu, P.C. We5t / 8050n1c 5tr1n95
U51n9 e4. (1.7) we f1nd that 0
Fr0m the rea11ty 0f P~(0) we f1nd that a ~ n = a~t. 7he a~5 c0mmutat10n re1at10n5 are
[a.",
:0,
[a~, a~,t ] = n3n, m3""
(1.14)
f0r n, m >1 0. 7he V1ra50r0 0perat0r5 [5] can 6e expre55ed 1n term5 0f the a,•5 6y +~
L.
E
~-
""
01m0Ln
-
m
(1.15)
where L 0 15 under5t00d t0 6e n0rma1 0rdered. 7hey 06ey the m0d1f1ed a19e6ra
[ L,, Lm1= ( n - rn)L,+ m + ~n(n 2 - 1 ) 3., m.
(1.16)
We f1nd 1n part1cu1ar that
L 0=
+ ~ a,,a,,"~t ,
1-~vP~22~0
(1.17)
m=1
L 1 = L* 1 =
" "1 -~0~0a
~ -"*,~ ~m~m+1
(1.18)
m=1
where
(0ff) 1/2 0 a~ = -1 -~
0x"
We may wr1te the 5tate 0f the 5tr1n9 1n 0CCupat10n num6er 6a515 1n term5 0f the creat10n 0perat0r5 a~* 6y +[x"(0)]=
(~0(x)+1A~a~t+h,,a~*a~*+;a"tA2+, 2 , ~}(x~(0)10).
(1.20)
7he vacuum 5at15f1e5 the e4uat10n a~1 (x~(0)10) = 0,
n>~1.
(1.21)
A1ternat1ve1y, 0ne c0u1d chan9e 6a515 t0 d1a90na112e x"(0), and th15 ha5 a1ready 6een c0n51dered 1n the 119ht-c0ne f0rmu1at10n 0f 1nteract1n9 5tr1n95 [3]. 7he vacuum
A. Neveu,P.C. We5t / 8050n1c5tr1n95
129
0f e4. (1.21) 15 0f the f0rm
=n1~11CneXP(-- 20t, XnXn~%J,) ~
(1.22)
7he act10n 0f the a,~* 0n (x~(0)10> pr0duce5 the we11-kn0wn c0mp1ete 5et 0f Herm1te p01yn0m1a15. 1n term5 0f c0mp0nent f1e1d5 we f1nd that e4. (1.10) ha5 a5 c0n5e4uence5
(1)
=
[ (1) 0 2--~
h~,. . . . . etc...
(1.23)
7 h e act10n 0f e4. (1.11) 15 0~A~ = 0 = V~1 0~h~, + A~2) = v~ 0~A~2) + h•.
(1.24)
We n0w w15h t0 f1nd an act10n wh1ch d0e5 n0t have the c0n5tra1nt5 0f• e4. (1.11) 6ut 1n5tead p055e55e5 9au9e 1nvar1ance5 that a110w the c0n5tra1nt5 0f e4. (1.11) t0 ar15e a5 a 9au9e ch01ce. C0n5e4uent1y, we expect an 1nf1n1te num6er 0f 9au9e 1nvar1ance5, 0ne f0r every 0ne 0f the c0n5tra1nt5 wh1ch 9enerate the c0nf0rma1 9r0up. 7h15 can 6e ach1eved ma55 1eve1 6y ma55 1eve1 6y 5ucce551ve1y re1ea51n9 the c0n5tra1nt5 0n ~k. At the f1r5t 1eve1, we re1ea5e the c0n5tra1nt L~k = 0 6ut ~p 15 5t1115u6ject t0 L 2 ~ = L~4~ = L 3 ~ = L2L1~k... = 0.
(1.25)
C0n51der n0w the 9au9e tran5f0rmat10n 0f 4J
9ap = L 1 A 1 ,
(1.26)
L 1 A 1 = L 2 A 1 = 0.
(1.27)
where A 1 15 5u6ject t0
U51n9 the f0rm 0f L• 1 91ven 1n e4. (1.18), we f1nd that th15 1nvar1ance c0nta1n5 the tran5f0rmat10n 8A1 = 0~,A1(x ) wh1ch 15 the a6e11an tran5f0rmat10n expected f0r a 11near12ed Yan9-M1115 the0ry. An act10n 1nvar1ant under 8~ = L•1A 115 91ven 6y ~ / ~ ( L 0 2 ~ -- 1 - 1 ~ L ~ 1 L 1 ) ~ p ) .
(1.28)
7he e4uat10n 0f m0t10n 15 91ven 6y (L 0-1-~L
1L1)~k=0.
(1.29)
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A. Neve~ P.C. We5t / 8050n1c 5tr1n95
Exp11c1t1y te5t1n9 th15, we f1nd that ( L 0 - 1 - •L•1L1)L•1A
1 = (L0L• 1 - L 1 - L-1L0)A1 = 0,
(1.30)
51nce L1A 1 = 0. E4. (1.29) wa5 pr06a61y kn0wn t0 a few pe0p1e 1n the 01d heyday 0f 5tr1n9 the0ry, 6ut ha5 6een red15c0vered m0re recent1y [8, 9]. 7he pr0ject0r P 0f a 5tr1n9 f1e1d 0nt0 the phy51ca1 5tate5 0f e45. (1.10) and (1.11) wa5 91ven 1n ref. [7]. At 10we5t 0rder 1t 15 91ven 6y 1 P = 1 - ~ L ~ 1 - ~ 0 L 1. (1.31) We n0te that at th15 1eve1 the e4uat10n 0f m0t10n 15 91ven 6y ( L 0 - 1)P~6 = 0.
(1.32)
1t ha5 6een pr0p05ed recent1y [9,10] that e4. (1.32) 15 the c0rrect e4uat10n 0f m0t10n 0f a11 1eve15 0f the 5tr1n9, and th15 5pecu1at10n ha5 6een enc0ura9ed 6y the a60ve c01nc1dence f0r the 5p1n 0ne at the f1r5t 1eve1. 1n ref5. [9] and [10], P ha5 6een f0rma11y c0mputed f0r a11 1eve15, and e4. (1.32) 15 exp11c1t1y n0n10ca1 at the 5ec0nd 1eve1 and m0re and m0re 50 f0r h19her 1eve15.0ne•5 5u5p1c10n5 are further ar0u5ed 6y the fact that P can 6e c0n5tructed 1n an ar61trary 5pace-t1me d1men510n, and that D = 26 15 n0t part1cu1ar1y fav0ured. 7he c1eare5t way t0 5h0w that e4. (1.32) 15 n0t the r19ht 9enera112at10n 15 t0 c0n51der the f1r5t 1eve1 0f the c105ed 6050n1c 0r1ented 5tr1n9. At th15 1eve1, the c0var1ant de9ree5 0f freed0m are de5cr16ed 6y a 51n91e 5ymmetr1c f1e1d h,,. 7he V1ra50r0 c0nd1t10n 1mp11e5 that 1t 5at15f1e5 02h~ = 0,
0~h~ = 0.
(1.33)
7he5e e4uat10n5 te11 u5 that at th15 1eve1 the c105ed 5tr1n9 c0nta1n5 0n1y a 5p1n tw0 and a 5p1n 2er0. 7he 9enera112at10n 0f e4. (1.32) f0r th15 1eve1 15 02RfRxh0x=0, where
(1.34)
Rf= (~f 0.00 --
02
)"
(1.35)
7he reader 1mmed1ate1y rec09n12e5 that th15 15 a n0n-10ca1 e4uat10n, wh1ch d0e5 n0t adm1t a ham11t0n1an f0rmu1at10n. Mak1n9 1t 10ca1 6y mu1t1p11cat10n 6y 0 2 1ead5 t0 add1t10na1 5tate5. 1n fact, 1t 15 kn0wn [11] that there 15 n0 L0rent2 1nvar1ant, 9au9e 1nvar1ant, 10ca1 act10n c0n5tructed fr0m h,, a10ne, wh1ch de5cr16e5 60th 5p1n tw0 and 5p1n 2er0. 7he
131
A. Ne0eu, P.C. We5t / 8050n1c5tr1n95
0n1y Way t0 aCh1eVe th15 15 t0 1ntr0dUCe an0ther 5Ca1ar f1e1d t0 de5Cr16e the 5p1n 2er0; th15 15 the We11-kn0wn E1n5te1n + ma551e55 5Ca1ar act10n. A5 We 5ha11 5ee, rather than the 5p1n 0ne, the f1r5t 1eVe10f the C105ed 5tr1n9 111U5trate5 the 9enera1 pattern, name1y, f0r a 10Ca1 f0rmU1at10n, the the0ry natura11y re4U1re5 5Upp1ementary f1e1d5, a5 We 5ha11 n0W dem0n5trate.
2. 6au9e c0var1ant f0rmu1at10n 0f the 11near12ed 5tr1n9 f1e1d the0ry 2.1. 0PEN 57R1N65 A5 0Ur 5tart1n9 p01nt, We take the act10n at the f1r5t 1eVe1, e4. (28) ~( ~ , ( L 0 - - 1 - - ~ L - 1 L 1 ) ~ 6
(2.1)
),
wh1ch 15 1nvar1ant under the 9au9e tran5f0rmat10n5 ~1~6=L~1A1,
(2.2)
8 2 ~ = L ~ 2 A 2,
where A1, A 2 and ~k are 5u6ject t0 L 1 A , = L 2 A 1 = 0,
L2~ = L2~ = L3~ = L 2 L 1 ~ . . . . .
0.
(2.3)
We 5ha11 n0w f1nd the c0var1ant f0rmu1at10n 1eve1 6y 1eve1, 5y5temat1ca11y re1ax1n9 c0nd1t10n5 (2.3). At the 5ec0nd 1eve1 1n 26 d1men510n5, the 0n-5he11 5tate5 are 0ne ma551ve pure ••5p1n tw0•• [ - a n 50(25) 5ymmetr1c trace1e55 ten50r]. 7he the0ry 0f wave e4uat10n5 f0r part1c1e5 0f a 91ven 5p1n 15 a c0n5e4uence 0f the the0ry 0f 1nduced repre5entat10n5 0f the P01ncar6 9r0up. 1t 15 a 5y5temat1c pr0cedure t0 f1nd the pr0ject10n 0perat0r5 0nt0 the phy51ca1 5tate5 0f a 91ven 5p1n. 1n the 5tudy 0f the5e 0perat0r5 0ne f1nd5 the we11-kn0wn wave e4uat10n5. 1n fact the part1c1e5 0f 5p1n 2er0, • and 1 are the except10n rather than the ru1e, 1n that mu1t1p1y1n9 the1r pr0ject10n 0perat0r5 6y 8 2 y1e1d5 a 10ca1 f1e1d e4uat10n. F0r 5p1n5 9reater than 0ne, mu1t1p11cat10n 6y 8 2 1ead5 t0 a n0n-10ca1 and 50 an unaccepta61e f1e1d e4uat10n. 7he pr0cedure t0 6e f0110wed 1n the5e ca5e5 wa5 w0rked 0ut 1n ref. [12] and 1t 1nv01ve5 the 1ntr0duct10n 0f 5upp1ementary f1e1d5. F0r examp1e ••pure 5p1n tw0•• 15 de5cr16ed 0n-5he11 6y the f1e1d h ~ = h ~ 5u6ject t0 the e4uat10n5
8~h~= h~ = ( 82-- m2)h~v=0.
(2.4)
7he pr0ject0r 15 we11 kn0wn [13]. 1t 1nv01ve5 term5 0f the f0rm 0p 0X0p0a//(82) 2 and hence mU1t1p11Cat10n (8 2) d0e5 n0t 1ead t0 a 10Ca1 f1e1d e4Uat10n. 1n fact, there 15 n0 Way t0 de5Cr16e 1n a L0rent2 C0Var1ant Way 0n1y a ma551Ve 5p1n tW0 part1C1e 1n term5 0f 0n1y h~v = hv~, 5U6jeCt t0 h~" = 0. 7he C0rreCt e4Uat10n5 0f m0t10n 1nV01Ve the
132
A. Neveu, P.C. We5t / 8050n1c 5tr1n95
1ntr0duct10n 0f a 5upp1ementary f1e1d ~ and are 91ven 6y
( - 0 2 + m2)h~, + ( 0~0°h0, + 0,00h0~ ) - 2 ~ 0 ° 0 x h 0 x D-2(
6~,
- D - 1
0 r 0 v~ -- ~ -
0~0"h~=
) 0 2~
,
0 2 - D -D 2m2) d~,
(2.5)
where D 15 the d1men510n 0f 5pace-t1me. 1ndeed, the5e c0up1ed e4uat10n5 1ead t0 the de51red re5u1t, name1y
0 = 3~h~,~= 4~= (0 2 - m2)h~,~.
(2.6)
7he 5tr1n9 f1e1d ~k[x"(0)] c0nta1n5 at the 5ec0nd 1eve1 h,, and A~2), [5ee e4. (1.20)]; h0wever, A (2) a5 we 5ha11 5ee, 15 9au9ed away and the a60ve d15cu5510n 1mp11e5 that we re4u1re 0ne extra 5upp1ementary f1e1d 4~t0 1mp1ement the 5p1n tw0 e4uat10n. 7h15 f1e1d 4~ w111 6e the 10we5t c0mp0nent 0f a 5upp1ementary 5tr1n9 f1e1d 4~(2)[x~(0)]. We n0w re1ea5e 50me 0f the c0n5tra1nt5 (2.3) 0f the f1r5t 1eve1, and ~k and 4~(2)are n0w 5u6ject t0 =
L1~(2) = L 2 ~ (2) . . . . .
. . . . .
0,
0.
(2.7)
7he m05t 9enera1 expre5510n 0f the c0rrect 0rder 15
(L 0 - 1 -1L~1L1-1~yL~2L2)1~-~- (L21--{-- 3/3L~ 2)t~(2) = 0,
(2.8)
( L~ + ~/3L2) 4J= ( aL 0 + 6)~ ~2).
(2.9)
7he u5e 0f the 5ame 13 1n (2.7) and (2.8) 15 re4u1red 6y demand1n9 that the e4uat10n5 0f m0t10n f0110w fr0m an act10n. An a1ternat1ve way 0f 5earch1n9 f0r a 9au9e 1nvar1ance 15 t0 demand that L 1 0n e4. (2.8) 5h0u1d van15h when we u5e e4. (2.9). Carry1n9 th15 0ut and u51n9 the c0n5tra1nt5 0f e4. (2.7), we f1nd
~L~1( 1 1 L 2 + 3yL2)~ + (4L~1L0 + 2L~1)67(2) + 9/3L~1d~(2)=0.
(2.10)
70 e11m1nate ~k 6y e4. (2.9) re4u1re5 y =/3 and we f1nd
•L•1(aL 0 + 6)ep <2)+ L•1(4L 0 + 2 + 9/3),p<2)= 0.
(2.11)
133
A. Neveu, P.C. We5t / 8050n1c 5tr1n95
Hence we c0nc1ude that a -- 8 and 6 = 4 + 9/3. App1y1n9 L 2 1n a 51m11ar way f1xe5 /3 = 1, and we f1nd the e4uat10n5 0f m0t10n
L0•
-
• n=1 ~ L • ,9/L , ) ,/, + (L 1 + 3 L
0,
( L 2 + 3L2) tp = (8L 0 + 13)4~(2) .
(2.12)
7h15 5y5tem 0f e4uat10n5 15 1n fact 1nvar1ant under the 9au9e tran5f0rmat10n5
~ p = L~xA1,
~1(~(2) =
~$2~6 = L • 2 A 2 ,
1LaA1,
(~2~(2)= 3 A 2 ,
(2.13)
w1th
L2A 1 = L2A1 . . . . .
0,
L1A 2 "~- L2A 2 .
0.
. . . .
(2.14)
We 5tre55 that e45. (2.12) are A 2 1nvar1ant 0n1y f0r D = 26. 1t may 6e p055161e, w1th the 1ntr0duct10n 0f further 5upp1ementary f1e1d5, t0 re1ax th15 c0nd1t10n. 7he c0rre5p0nd1n9 act10n 15 91ven 6y
•( tp, ( L 0 - 1 - ~ L ~ 1 L 1 - 1 L ~ 2L2)~p ) +( ~,( L2~1 + 3L~ 2)4,(2)) -~(ep(2),(8L0 + 13)e~(2)).
(2.15)
7h15 c0mp1ete5 the ana1y515 0f the 5ec0nd 1eve1, and we n0w 90 t0 the th1rd and f0urth 1eve15. At the th1rd 1eve1, we have a5 0n-5he11 5tate5, 0ne ma551ve vect0r, 0ne ant15ymmetr1c 5ec0nd rank ten50r and 0ne • pure 5p1n-three••. 7he 1atter 15 1n effect de5cr16ed 6y the f1e1d ~ p = 4~(~p) 5u6ject t0 8~4~ 0 = 0 = 4~p. 7h15 re4u1re5 the 5upp1ementary f1e1d5 4~ and 4~ t0 de5cr16e 1t5 pr0Pa9at10n. 7he e~ f1e1d 15 5upp11ed at th15 1eve1 6y the 5ec0nd c0mp0nent f1e1d 0f e~(2), 1.e., 4~(2)[x~(0)] -- [4~(2)(x) + a.~*a, .~ ~ , + • • • ] (x~(0)10).
(2.16)
Hence, we 0n1y re4u1re 0ne new 5upp1ementary 5tr1n9 f1e1d 4,(3)[x~(0)] 51nce the ant15ymmetr1c ten50r and the vect0r d0 n0t re4u1re 5upp1ementary f1e1d5. 0 n e can carry 0ut the c0unt1n9 0f 5tate5 at the f0urth 1eve1 and 0ne f1nd5 that 0ne new 5upp1ementary 5tr1n9 f1e1d 15 re4u1red ~(4)[x"(0)].
134
A. Net)e•u, P.C. We5t / 8050n1c 5tr1n95
At the f0urth 1eve1 the f1e1d5 are 5u6ject t0
0 = L54J = L ~ = L4L1~6... etc.,
0 = L3~6(2)= L2Laep~2)= L3ep~2)... etc., 0 = L246(3)= L2~6(3)... etC., 0 = L14~4)... etc.
(2.17)
7 h e act10n 15 06ta1ned a5 f0r the 5ec0nd 1eve1; 0ne wr1te5 d0wn a11 p055161e term5 0f the c0rrect 0rder and re4u1re5 the re5u1t t0 6e 9au9e 1nvar1ant. 7he act10n 15 f0und t0 6e
((
21 ~
41
L0--1--E-2-~L-nLn n=1
4-(~,(L
~d
"4- (~,(L214-3L~2)4~ (2))
2L•14- 8L 3)4~(3)) 4• ( ~ , ( L ~ 3 L
14- ~ L ~ 4)~)(4))
---12(4~(2),(8L 04- 134- 2L•1L 1 - - L 2L2)~ (2)) 1(~6(3), (8L0 4• ~)4~(3)) -- 1(4~(4), (12L ° + 2~7)4~(4)) -- 3( L2* (2), ,(4, ) -- 6( L14~(2), ~(3)) -- 2( L2* (2), L14~(3)) -- 10( L1~ (3), 4~(4)). (2.18) 1t 15 1nvar1ant under the 9au9e tran5f0rmat10n5 1
311p = L•1A1,
81¢~(n+1) = ~ n L . A 1 ,
32~k=L 2A2,
32ep~2)=3A2+~L~2L2A 2,
1 1 324~0) = ~L1A 2 - 9L• xL2A 2,
62d•(4) = ~L2A 2 .
(2.19)
Further 9au9e tran5f0rmat10n5 can 6e f0und 6y tak1n9 5pec1a1 va1ue5 0f A 1 and A 2, 5ay A 1 = + L 2A 3,A 2= -L•1A3 wh1ch 91ve5 8~6= L 3A 3 and expre5510n5 f0r 3ff~"). 51nce the 11near12ed act10n (2.18) 15 4uadrat1c 1n the 5upp1ementary f1e1d5, 0ne c0u1d 1n pr1nc1p1e 1nte9rate them 0ut, rec0ver1n9, up t0 that 1eve1, the pr0ject0r 0f ref. [7]. 1n 0rder t0 d0 th15, h0wever, 0ne mu5t 1nvert the matr1x 0f the 4uadrat1c f0rm 1n the 5upp1ementary f1e1d5, a pr0ce55 wh1ch 15 a1ready very pa1nfu1 at f0urth 1eve1.
135
A. Neveu, P.C. We5t / 8050n1c5tr1n95
8ef0re c0nt1nu1n9 t0 the f1fth 1eve1, 1t 15 u5efu1 t0 ref0rmu1ate the re5u1t at the f0urth 1eve1 6y 1ntr0duc1n9 a further 5upp1ementary f1e1d. 7h15 a150 a110w5 u5 t0 91ve 50me 5ect0r5 0f the act10n t0 a11 1eve15. 7he ~k e4uat10n 0f m0t10n 15 0f the f0rm
L 0 - 1 - )-•. ~ n L~ , L , + + 5upp1ementary f1e1d term5 = 0.
(2.20)
n=1 7h15 f0rm 15 nece55ary 1n 0rder t0 e11m1nate the LmL 0 term5 up0n the mu1t1p11cat10n 6y L,.. Act1n9 w1th L 1 we f1nd the re5u1t
1
.+2
]
n=1 ~ n L - n L1Ln "•[•- "-n-~1~1Ln +1 1~ -•[- 5Upp1ementary f1e1d term5
0. (2.21)
1n 0rder t0 e11m1nate the + term we re4u1re 5upp1ementary f1e1d5 ~(n+1) wh05e e4uat10n5 0f m0t10n are n+2 ) L1L,, + n n - - - ~ L , + 1 + = 5upp1ementary f1e1d5.
(2.22)
F0r the act10n 0f L2 0n e4. (2.20), we f1nd the re5u1t 3 ~( L12 + ~L2)~6~
8 L n (L2Ln+n ~L~1(L1L2+~L3)~ ~ __~_n
n=2
n+ n+
+ 5upp1ementary f1e1d term5 = 0.
(2.23)
We 065erve that the f1r5t tw0 term5 1nv01ve the 5ame 0perat0r5 a5 0ccur f0r the app11cat10n 0f Lp 1n fact, 0ne can ehm1nate 4~ fr0m th15 e4uat10n u51n9 the 1dent1ty
n(n+4) L2 L n -1
n+2
2 2
(
Ln+2=~Ln(L1+ 3L2)-2L 1 L1Ln+n
2 n2-2n-2[ 3 n+1
n+2 ) Ln+1 n+1
(n+1)(n+3) ] L1L"+1 -~
n+2
Ln+ 2 . (2.24)
H0wever, 1t 15 c0nven1ent t0 1ntr0duce a further 5et 0f 5upp1ementary f1e1d5 X(") (n >1 4), the e4uat10n 0f m0t10n f0r Xt") 6e1n9 n2-4 L2Ln - 2 +
n
) L. ~6= 5upp1ementary f1e1d term5.
(2.25)
136
A. Neveu, P.C. We5t / 8050n1c 5tr1n95
7he5e e4uat10n5 f0110w fr0m the act10n
• ~p, L 0 - 1 -
, L . ~p +
,=1 2n
~, L ,L 1 + n -
n=1
n+ 1
L n-1 -
n(n+4,L ) ) n+ 2
n-2 X(•+2)
(~(n+1)
(2.26)
p1u5 5upp1ementary term5 a10ne. Vary1n9 + we f1nd the ~p e4uat10n t0 6e 0f the f0rm
L0-1-
+
n~1
-~nL . L . ~,+
n=1
L ,L 1 + n -
n + 1
n
L ~ , L ~ 2 + n - ~ - - 2 L ~ , ~ 2 X(~+2)= 0.
1
(2.27)
Fr0m th15 e4uat10n we may read 0ff the A 1 and A 2 9au9e tran5f0rmat10n5 t0 a11 0rder5. 5u65t1tut1n9 ~p = L ~ A ~ , we f1nd 1
~1~ =
L1A1,
81~(n+1) =
"~nL.A1,
3~X(n) = 0.
(2.28)
Wh11e 5u65t1tut1n9 3~6 = L 2A 2, we f1nd ~2t~ (2) "~- 3A 2,
32~0) = •L1A 2, 1
~2(~ ( n ) = 0,
f0r n >1 4,
~X (n+2) =
~nLnA2 .
(2.29)
We c0u1d further 51mpf1fy the a60ve f0rmu1ae 6y 1ntr0duc1n9 tw0 5upp1ementary f1e1d5 f0r each 0f the expre5510n5 ( L 2 + 3(L2))~ and (L1L 2 + 8(L3))+. 7h15 c0rre5p0nd5 t0 X (2), X (3) f0r the act10n 0f L 2 and ~(2) and (/)(3) f0r the act10n 0f L 1. 7he act10n, h0wever, w111 0n1y c0nta1n the c0m61nat10n5 0 (2) + X (2) and 4~(3)+X(3). 7h15 fact c0u1d 6e enf0rced 6y demand1n9 that the act10n have the 5ymmetr1e5 ~(2) •••) ~(2) •4• ~(2),
X (2) ~ X (2) • ~ (2),
4~(3)~ ~(3) + ,4(3),
X0) ~ X(3) • f/(3).
(2.30)
A. Ne0e~ P.C. We5t / 8050n1c 5tr1n95
137
1n th15 ca5e the re5u1t1n9 9au9e tran5f0rmat10n5 are 1
~1~(n+ 1)= .~n LnA1 ~
814~= L•1A1, ~xX~") = 0, 814 = L• 2A 2,
32~ (n) = 0 ,
1
~2X~"+2) = ~n L, A 2,
n>~1, (2.31)
82X ~2)= ~A 2.
Hencef0rth we w111re9ard X(2) and X(3) a5 6e1n9 a65ent 0r 9au9ed away 6y ~(2) and f20). 1t 15 1n5truct1ve t0 c0mp1ete the 54uare5 1n the act10n 0f e4. (2.26). 1t 6ec0me5 0~
~(4~, ( L 0 - 1)4~) - • ~
1
- - ( 6 ~ " ) , 6~")) + 5upp1ementary f1e1d5 term5, (2.32)
n ~ 1 21•/
where 6 ( n ) = Lnt~ -- 2 ( n 2 -- 1) dp(n) -- 2 n L - 1 ~ ( n + 1 ) -
2 r t L - 2 X (n+2) -- 2 ( n 2 -- 4 ) X ( n ) .
(2.33)
Under A 1 a11 the 6 (") are 1nert except 6 ~1), wh11e under A 2 they tran5f0rm a5 826(1) =
L2L1A2,
826(3) = --
~26 ~") = 0
~26(2)= 4 ( L 0 + 1)A 2 - 2 L 1 L 1 A 2 ,
3L1A 1, f0r n >1 4.
(2.34)
An0ther u5efu1 c0m61nat10n 15 ~-~,+3) = n (n + 2)4~("+3) + (n + 1)L1~ ~"+2) - L , + 145~1) + 3(n + 1)X(n+3) . (2.35) 7h15 15 a150 1nert under A 1 and tran5f0rm5 under A 2 a5
~2~~4)= L~A 2,
32~") = 0,
n >1-5.
(2.36)
138
•4. Neveu, P.C. We5t / 8050n1c5tr1n95
1t 15 n0W a 51mp1e matter t0 Ver1fy the 1nVar1anCe 0f the f0110w1n9 act10n at f0Urth 1eVe1 Under the A (1) and A (2) 9aU9e 1nVar1anCe5 4
1(~p,(L 0 - 1 ) ~ p ) - ~ E
1
~---n(6("), 6("))
n=1
- 2(~6(2), (L0 + 1)d,~(2)) -1-1(~(4), ~(4)) -- 8(X(4), ( L 0 4- 3 ) X ( 4 ) ) .
(2.37)
7he reader may w0nder what the re1at10n 15 6etween the a60ve f0urth 1eve1 act10n and that 91ven prev10u51y 1n e4. (2.18). 1n fact, X(4) can 6e 9au9ed away 51nce the a60ve act10n (2.37) p055e55e5 the add1t10na1 5ymmetry 34~(2) = - 2L • 2 ~ (4) , 3~(4) = -- ~ ( 4 )
8~ (3) = 2L• a12(4), ~(4) = 382(4).
(2.38)
Up0n 9au91n9 away X (4) w e 06ta1n the prev10u5 act10n 0f e4. (2.18) and the prev10u5 tran5f0rmat10n5 1aw5 0f e4. (2.19) pr0v1ded we make the nece55ary c0mpen5at1n9 tran5f0rmat10n5. Hav1n9 6u11t up the a60ve f0rma115m, the f1fth 1eve1 15 ea511y f0und. 7he act10n 0f e4. (2.37) 15 A 1 and A 2 1nvar1ant pr0v1ded we add the term5 12 ( X (4), L•1, X(5)) + 9(X (5), X (5)) + 4(L1X (4), L1X(4)). 7he 0n1yA 1 and A 2 1nVar1ant term we may add t0 the act10n 0f e4. (2.37) 15 c (~.(5), ~.(5)),
(2.39)
a5 we11 a5 extend1n9 the 5um 0f the 6n~5 fr0m 4 t0 5 . 7 h e c0eff1c1ent c 0f th15 new term 15 ea511y f0und 6y demand1n9 that the ~2(4) 5ymmetry 1nher1t5 t0 the f1fth 1eve1. 0 n e f1nd5 that c = + 2. 7 h e f1fth 1eve1 then ha5 a new 5ymmetry wh1ch 15 91ven 6y 3~ (2) = - 2L• 312(5), 8 ~ (4) = 2 L • 1~ (5),
3X (4) = 0,
3~ (3) = 0, 34~(5)= •f2 (5),
3X (5) = 3~2(5) ,
(2.40)
wh1ch a110w5 0ne t0 9au9e away X (5). 7h15 5ymmetry /2 (5) and that 0f ~(4) a r e an 1nvar1ance 0f the ~p e4uat10n 0f m0t10n due t0 the 1dent1ty (2.24). 7he 51xth 1eve1 15 pre5ented 1n append1x A. We w111 n0w dem0n5trate that we can rec0ver e45. (1.10) and (1.11) 6y a 5er1e5 0f 9au9e ch01ce5. 0 f c0ur5e, 51nce 0ne ha5 an act10n 1nvar1ant under ar61trary 9au9e tran5f0rmat10n5, we mu5t ch005e a very 5pec1f1c 9au9e 1n 0rder t0 arr1ve at e45.
139
A. Neveu, P.C. We5t / 8050n1c 5tr1n95
(1.10) and (1.11). F1r5t1y, We 9aU9e X(") = 0 U51n9 12("); We then U5e A (2), A (3), A(4) ... t0 9aU9e away 4~(2),4~0), ~(4) . . . . Up0n mak1n9 the nece55ary c0mpen5at1n9 tran5f0rmat10n5, the re5u1t1n9 e4uat10n5 w1th 4,(2) = ff0) = 4,(4) = 0 are 5t111 1nvar1ant under A 0). We may u5e A (t) t0 5et Lt~p = 0. 5u65t1tut1n9 Lt~p = d~(2) = d~(3) = 4,(4) . . . . . 0, we rec0ver e45. (1.10) and (1.11). 7h15 dem0n5trate5 that the act10n c0n5tructed a60ve d0e5 1ndeed de5cr16e the c0rrect 5pectrum f0r the 0pen 6050n1c 5tr1n9. 7h15 pr0cedure 15 50mewhat 5u6t11e and make5 u5e 0f 0n-5he11 9au9e 1nvar1ance5 1n a 51m11ar way t0 the way 1n wh1ch 0ne dem0n5trate5 that QED de5cr16e5 a ph0t0n w1th 2 de9ree5 0f freed0m. 7he reader may ver1fy, f0r h1m5e1f the a60ve 5tatement. 2.2. CL05ED 57R1N65 7 h e c105ed 5tr1n9 x ~ ( 0 ) can 6e wr1tten 1n the f0rm +00 E e1n°Xn9
X9( 0 ) =
~t ~
where x " = x "
(2.41)
--00
We def1ne the creat10n and de5truct10n 0perat0r5 6y 3
P.(0)
1
t 3x~,
-
=
1
0x ~
--=
1~Xt~ 2rra• 00
+00
~ ~., •/r(2a•) 1/2 n=~-00 -1n, t*
2~ra~ 0 0 1
P~(0)
0x ~
---+----=
=
1
+ ~0
..
rr(2a•) 1/2 n=~00 e-1~"a"
(2.42)
7 h e V1ra50r0 0perat0r5 are 91ven 6y L.=
• 1 r a ~f ]
d0e-1"°P~
( )02
+00
= ~1 m ~
£ , = ~ra~
f~
d0e/"°
- - •R
~ ~(
0) 2 =
~, m
..
E
a~a,~
m,
- - ~
+00
-~-~
09 m 0~ n ~
--
-- rn
"
(2.43)
c10
7 h e a•5 5at15fy the re1at10n5
[~2,~*] =n~.,m~
[ ~:., ~*] = .an, m~ , [am,a,]
f0r n , m > ~ 1 ,
=0
(2.44)
1t 15 we11 kn0wn that the 0n-5he11 5tate5 0f the 6050n1c 0r1ented c105ed 5tr1n9 are de5cr16ed 6y the e4uat10n5 ( L 0 + 2 0 - 2 ) + = 0, ( L 0 - L0)4* = 0,
L,~6 = L,,~k = 0,
n >1 1.
(2.45)
140
A. Neveu, P.C. We5t / 8050n1c 5tr1n95
At the f1r5t 1eve1, the5e e4uat10n5 have a ma551e55 ••pure 5p1n 2•• and 0ne 5p1n 2er0. A5 d15cu55ed 1n the 1ntr0duct10n, the5e 5tate5 can 0n1y 6e de5cr16ed 1n a 10ca1, 9au9eand L0rent2-c0var1ant way 6y the f1e1d5 h,~ and 4~. 51nce the 5tr1n9 f1e1d ~ 0n1y c0nta1n5 h ~ at the f1r5t 1eve1, we mu5t add a 5upp1ementary 5tr1n9 f1e1d ~(1). 7he f1r5t 1eve1 re5u1t 15 f0und 1n much the 5ame way a5 f0r the 0pen 5tr1n9; we 6e91n 6y re1ax1n9 50me 0f the c0n5tra1nt5 0n 4~, wh1ch 15 n0w 5u6ject t0 (2.46)
(L0 - L0)~ = L~k = L2~ = 212~k= L2~P= 0 a5 we11 a5
L1~6(1)= /7-,1~(1)=0. 7he act10n 15 f0und t0 6e ~(4~, ( L 0 + L0 - 2 - L
,L 1
L~1L1)~) "~-(1p,L
-
1L~1~ (1))
~(~(1) (L0..1~20)~(1))
(2.47)
and 15 1nvar1ant under the tran5f0rmat10n5
~1~k= L-1A1 4- L-1A1,
~1 dp(1) = L1A1 + L1A1•
(2.48)
m
where the 9au9e parameter5 A 1 and A~ are 5u6ject t0 ( L 0 - L 0 + 1)A1= 0,
( L 0 - L 0 - 1)A1= 0,
L1A 1=0, L2A 1 = L1A 1 =
L,A 1=0, 0,
L2A 1 = L1A 1 =
0.
(2.49)
7he act10n 0f e4. (2.47) at the f1r5t ma55 1eve1 can 6e eva1uated 1n term5 0f the c0mp0nent f1e1d5 h ~ and 4~, and 15 n0ne 0ther, m0du10 a tr1v1a1 f1e1d redef1n1t10n, than the 11near12ed E1n5te1n p1u5 5ca1ar act10n. 7he treatment 0f h19her 1eve15pr0ceed5 1n e55ent1a11y the 5ame way a5 f0r the 0pen 5tr1n9. We treat here 0n1y the next (5ec0nd) 1eve1, where the • 5p1n•• can 6e a5 h19h a5 f0ur. 0 n e f1r5t arran9e5 the A1, A1 9au9e 1nvar1ance5, re1ax1n9 appr0pr1ate1y the c0nd1t10n5 (2.49). 0ne then c0n5truct5 an act10n 1nvar1ant under Ap A~, 1ntr0duc1n9 the nece55ary 5upp1ementary f1e1d5. After that, 0ne 1mp05e5 the new A 2, -42 1nvar1ance5. 7he c0nd1t10n5 (2.49) are n0w rep1aced 6y 0 = L2a1 = L
51 = E 2 A 1 = E1 A1 = L15
= E A2 = c 7,2 =
(2.50)
Up0n app1y1n9 th15 pr0cedure, 0ne f1nd5 that f1ve new 5upp1ementary f1e1d5
141
A. Neveu, P.C. We5t / 8050n1c 5tr1n95 -
•
(02, 02, 02, X 2, X 2) are needed at the 5ec0nd 1eve1, w1th tran5f0rmat10n 1aw5 3102 = L1A1,
~1~2 = L1A1,
~1X2 ~---L 2 A 1 ,
~10~2 = 0,
(~202
: 3A 2•
L2A1,
820=L~2A2+2~2A2,
~2~2 = 3A 2,
~2X 2 = 2 L 1 A 2•
~1X2 =
~2X2 = 2L1A2, (2.51)
~20~2= L2A 2 + L2A 2"
At th15 1eve1, the 1nvar1ant act10n 15 ~(4~, ( L 0 + E 0 - 2
- L•1L
1
1 - L•1L
1 - •L
~(01,(L0~{~-J7-~0)01)~-~(~,L
+(~,(2~-1+~2
•
2L2 - 1J~,~ 2L2) 1~) --}-(1/J, L•1L 1
101)
t
2L 2 0 ~ ) - 7 ( 0 2 , ( L 0 + L 0 + 2 ) 0 ~ ) •
2)~2)+~(~,L 12 212)+~(~,2 1L 2x2)
-(01, L 12102)- (01, 2 1L1~2)- ( ~ , ( L0 + ~0 + ~)~2) 1 -1 + ~(0~, L~1L1~2) ~r~,~,2 L 112)- (02,( L0 + L0 + ~)0~) + ~(02, L 1L10~) 5 1 L - - 2 ( 0 2 , 1-~-1X2)--4(X2,( 0~-1~-~04-1)X2) ~(X2, L - 1 L 1 f ( 2 ) - •(X2, (L0 + L0 + 1)X2) 1 - ~(x2,
L~1L1x~)
-
3 • ~(0~, L202 + L2~2) + ~ ( ~ L ~ 1
+1(X2L~1-02L~2,21X2-L202)-~(0~2,(L0+
- 0 2 2 ~ 2, L122 - L~02) L 0 + 2)0~2).
(2.52)
3. 1nteract10n5
1n th15 5ect10n, we w111dem0n5trate h0w t0 c0n5truct the 1nteract10n5 f0r the 0pen 6050n1c 5tr1n9 and carry 0ut the f1r5t 5tep 0f th15 pr0cedure. 70 f1nd the 1nteract1n9 the0ry fr0m 1t5 11near12ed f0rm, we w111 emp10y a 9enera112at10n 0f the N0ether meth0d. 1n th15 meth0d, 0ne 5tart5 w1th the 11near12ed the0ry wh1ch p055e55e5 a6e11an 10ca1 1nvar1ance5 and a 51m11arnum6er 0f r191d 1nvar1ance5. 0 n e then make5 the r191d 1nvar1ance5 10ca1 and f1nd5 an 1nvar1ant n0n-11near act10n 0rder 6y 0rder 1n the c0up11n9 c0n5tant. 7h15 15 ach1eved 6y kn1tt1n9 t09ether the r191d and 10ca1 a6e11an 1nvar1ance5. F0r the 11near12ed the0ry 91ven 1n the prev10u5 5ect10n, the 11near12ed a6e11an 1nvar1ance5 are the tran5f0rmat10n5 w1th parameter5 A(1), A(2), A(3), etc. 7he r191d 1nvar1ance5 are the u5ua1 r191d r0tat10n5 0f the f1e1d5 under the 9r0up 6 . 1n fact,
142
A. Neveu, P.C. We5t / 8050n1e 5tr1n95
there are an 1nf1n1te 5et 0f 5uch 1nvar1ance5, 51nce we can make an 1ndependent r0tat10n at every ma55 1eve1. 7he f1r5t 5uch 1nvar1ance 15 the 0ne wh1ch r0tate5 a11 ma55 1eve15 1n the 5ame way; 1t 15 91ven 6y
~ = [~-~(1),~],
(3.1)
x~(0). Let u5 den0te the nth r191d r0tat10n 6y t2(n). t2
(3.2) where 7"1• and 72 are 0perat0r5 0f the f0rm
=
(3.3)
c)100)105)10c)
wh1ch are t0 6e determ1ned. 7he 1a6e15 a, 6 and c c0rre5p0nd t0 the three ••1e95•• 0f 7~ wh1ch are 111u5trated 1n f19. 1: 7here a150 ex15t 51m11ar e4uat10n5 f0r the 5upp1ementary f1e1d5 [4~(~)). 1n th15 5en5e, the meth0d we are u51n9 repre5ent5 a 9enera112at10n 0f the u5ua1 N0ether meth0d. Up0n 5u65t1tut1n9 the n0w-10ca1 var1at10n5 0f ]~p) 0f e4. (3.2) and th05e f0r ]4~(n)) 1nt0 the 11near12ed act10n, we w11106ta1n an expre5510n wh1ch 15 11near 1n (t2 (n) ] and 6111near 1n [4J) and J~(")). 7he act10n 0f the L,•5 wh1ch make up the 11near12ed act10n d0e5 n0t, 1n 9enera1, 06ey the L1e6n1t2 ru1e. Hence the 7•5 w111 have t0 6e very 5pec1a1 0perat0r5, 50 that the act10n 0f the L,•5 pr0duce5 further L,•5 0n the 0ther 1e95. 1n the d15cu5510n wh1ch f0110w5, we w111 0n1y refer t0 t2(1), 6ut the exten510n t0 t2 (~) 15 1n pr1nc1p1e c1ear. 7he var1at10n 0f the 11near12ed act10n can then 6e wr1tten 1n the f0rm
(~2(x) 1L.8 ~) ,
61~ua>=a
72•
1Q6 >
1~c> F19. 1.
+ a~
(3.4)
14J6> 1Qc>
A. Neveu, P.C. We5t / 8050n1c 5tr1n95
143
where 5 (•) 15 an expre5510n 6111near 1n 14~) and 14)(•)). 7h15 var1at10n 0f the 11near12ed act10n can 6e cance11ed 6y the add1t10n 0f a term 0f 0rder 9 t0 the act10n. 7he var1at10n 0f th15 term at 0rder 90 re5u1t5 fr0m 0n1y the var1at10n 0f the f1e1d5 under the a6e11an 1nvar1ance5 wh05e parameter A(1) 15 1dent1f1ed 6y 1
1A(~)> = -1~2(x)). 9
(3.5)
($20) 1L15(1)
(3.6)
F0r examp1e, a term 0f the f0rm
15 cance11ed 6y the add1t10n 0f the term 0f the f0rm 9{4J15 (1) .
(3.7)
7he var1at10n5 0f th15 new act10n at 0rder 9 are then cance11ed 6y add1n9 new term5 t0 the act10n and p055161y new term5 t0 the tran5f0rmat10n 1aw5. 7h15 15 repeated unt11 0ne 06ta1n5 an act10n 1nvar1ant t0 a11 0rder5 1n 9. Fr0m the 119ht-c0ne f0rmu1at10n 0f the 1nteract1n9 5tr1n9 f1e1d the0ry and the 9e0metr1ca1 1nterpretat10n 0f 1nteract10n5 5uch a5 5p11tt1n9, j01n1n9 and rearran9ement 0f 5tr1n95, 0ne can expect at m05t 4uart1c 1nteract10n5 f0r 0pen 5tr1n95 and cu61c 0ne5 f0r c105ed 5tr1n95 a150 1n a 9au9e c0var1ant f0rmu1at10n. H0wever, the vert1ce5 w111 nece55ar11y 1nv01ve the 5upp1ementary f1e1d5 4~(~). 7he 0ccurrence 0f 1nteract10n5 at 1ea5t 4uadrat1c 1n ~(~) re5u1t5 1n the pract1ca1 1na6111ty t0 1nte9rate 0ut the 5upp1ementary f1e1d51n the path 1nte9ra1. 5uch term5 are 1nev1ta61e; f0r examp1e, there ex15t5 1n the c105ed 5tr1n9 5ect0r the u5ua1 9rav1t0n-ma551e55 5ca1ar c0up11n9, even 6ef0re tak1n9 the f1e1d the0ry 11m1t, hence a n0n-2er0 d~(1)~ (1) 1nteract10n term. 7h15 5h0w5 that the n0n-10ca1 f0rmu1at10n 0f the free 5tr1n9 the0ry 0f ref5. [9] and [10] can def1n1te1y n0t 6e 9enera112ed t0 the 1nteract1n9 ca5e. 7he a60ve pr0cedure can 6e carr1ed 0ut 1eve1 6y 1eve1, and we w111 n0w f1nd the 0rder 9 1nteract10n5 at the f1r5t 1eve1. We 6e91n w1th the act10n
1-•L
(3.8)
U51n9 the var1at10n5 0f 14J) 0f e4. (3.2), we f1nd (4•61(L0 - 1 - ~L-~L1)6{(4,a1(~2c171 + <~2.1 (~c1 72~ } •
(3.9)
7h15 mu5t 6e expre55161e 1n the f0rm ($2a1L~(4%1 (4%1W,
(3.10)
144
A. Ne0eu, P.C. We5t / 8050n1c 5tr1n95
wh1ch we can cance1 w1th the var1at10n 0f the term 99( ~1(~k6[ (~P¢1W.
(3.11)
C0mpar1n9 e45. (3.9) and (3.10), we re4u1re the 0perat0r5 7, and W t0 5at15fy
L~W= (L 0 - 1 -
1L•1L1)671
-~- ( L 0 - 1 - 1 L ~ 1 L 1 ) c 7 2 .
(3.12)
L0n9 a90, a cyc11ca11y 5ymmetr1c three-p01nt vertex wa5 wr1tten d0wn [14] 1n the f0rm
(-1)mY(n)
V = exp { - n=1 (01a-n016-m-]- 016-n0Lc m~1- 01t~-n0£a-m) 1~(m ~ - 1 ~ F ~
}
2~1 -}- 1) 10) "
m=0
(3.13) 1t 15 1n5truct1ve t0 eva1uate V, tak1n9 the externa1 5tate5 t0 c0nta1n 0n1y the ma551e55 Yan9-M1115 vect0r5; 0ne f1nd5 the re5u1t k1" E3e1" e2 +
k2"
E1E2"E3 -1- k 3 " e2e1" 83 "4- k1
" E3k2 " e1k3" E2-
(3.14)
7he 1a5t term, 0f c0ur5e, d15appear5 1n the 2er0 510pe 11m1t. 7h15 vertex d1ffer5 fr0m 0ne wh1ch 15 ant15ymmetr1c under the 1nterchan9e 0f any tw0 externa1 11ne5 6y 10n91tud1na1 term5 wh1ch are: E~E~[~k~p
+ 1~
10
,~1,~3,~2
+
1~
~ 0
(3.15) We theref0re take 0ur 0perat0r5 t0 6e 0f the f0rm
W ~ [1-Jr0t(La-1 + L6-1 + Lc1) +f1( L6-1Lc-1 + L~-1L6-1 d- U -1 L ~-1] 4- 8L ~ 1L6 1 L c 1 ]
V,
71= Y(1+ a•Lc1)V, (3.16)
72 = E(1 "~-0L~( L6 1 + Lc 1 + La-1)) V"
7he5e are the m05t 9enera1 0perat0r5, tak1n9 1nt0 acc0unt the c0n5tra1nt5 f0r the f1r5t 1eve1 wr1tten 1n the prev10u5 5ect10n: L 11~)
=
L21~k) = L~6p) . . . . .
0.
(3.17)
A. Neveu, P.C. We5t / 8050n1c5tr1n95
145
( L ~ - L~ + L6~1- L6 + L ~ ) V = 0 ,
(3.18)
U51n9 the re5u1t [15]
and the ana1090u5 re5u1t5 06ta1ned 6y cyc11ca11ypermut1n9 a, 6 and c, we can eva1uate L~ 0n W and f1x the a60ve c0eff1c1ent5. 0ne f1nd5 W=[1
--
1
a
L 6
2(L-1+-1
+ Lc•1)
1[ La--1 L 6- 1 + L 6--1 L c--1 + L c- 1 L ~- 1 ]~- - ~L ~ 1L 6--1 L ~--1J] V , 2
-•[- 2~
71=V, 72= --(1--(L6•1+La1+Lc•1))V
,
(3.19)
where 7~(6, a, c) = 71(c, 6, a ) .
(3.20)
At th15 1eve1, 72• happen5 t0 6e cyc11ca11y5ymmetr1c, and the permutat10n (3.20) 15 1rre1evant. 1t may 6ec0me re1evant 1n next 0rder. Hence the 0rder 9 re5u1t at the f1r5t 1eve115 ~(#1L 0 - 1 -
~9<~k~1<#61 <~kc1W.
(3.21)
11a26> + <61
(3.22)
~L1L 16k5 -
7h15 act10n 15 1nvar1ant t0 0rder 90 under 1
~1~6>
=-L6 9
-
1n the 11terature 0f f1fteen year5 a90 0n dua1 m0de15, there appeared 0ther three-Re99e0n vert1ce5. 0ne, 1,V, 15 ju5t W w1th the 0pp051te cyc11c 0rder1n9 0f externa1 11ne5. 80th W and W were 06ta1ned fr0m the 5c1ut0 [16] vertex 5 6y app1y1n9 appr0pr1ate tw15t 0perat0r5 12t~5t, acc0rd1n9 t0 f19. 2 and ref5. [14] and [15]. 1t ha5 6een 5h0wn 1n ref. [17] that 0ne 06ta1n5 1n the 2er0-510pe 11m1t the c0rrect Feynman d1a9ram5 0f a f1e1d the0ry 6y 5umm1n9 0ver tw15ted and untw15ted 1nterna1 11ne5 0f the 5tr1n9 d1a9ram. 7h15 re5u1t 5h0w5 that 0ne 5h0u1d 1nc1ude a11 f0ur vert1ce5 W, 1~, 5, 5 0f f19. 2 (5u1ta61y m0d1f1ed 6y Ln~5 1n the manner 5h0wn a60ve f0r W) 1n the act10n and tran5f0rmat10n 1aw5. 7he re1at1ve c0eff1c1ent5 mu5t 6e determ1ned 6y c105ure 0f the a19e6ra 0f 9au9e tran5f0rmat10n5. Fr0m (3.22) and (3.19), 1t 15 06v10u5 that the 5tructure c0n5tant5 0f th15 a19e6ra are e55ent1a11y 91ven 6y the matr1x e1ement5 0f the three-re99e0n vert1ce5. 7h15 w1116e dea1t w1th 1n a future pu611cat10n.
146
A. Neveu, P.C. We5t / 8050n1c 5tr1n95
//
~
U
= a
~C 6
N
Q~w15t 5 ~t"
~c
F19. 2. 4. 0ut100k 50me n0ve1 4ua11tat1ve feature5 emer9e fr0m the ana1y515 0f the prev10u5 5ect10n wh1ch we expect t0 carry t0 a11 0rder5 1n 5p1te 0f techn1ca1 c0mp11cat10n5. 7he5e feature5 are 6e5t 5een and exp1a1ned 6y 901n9 t0 funct10na1 f0rm. 1n 5uch a f0rm, under A ~"), the 9au9e tran5f0rmat10n 0f ~k may 6e wr1tten a5
f 47 d0e1"0p2(°) n=1 -~
=~a~p2(0)Ad0,
(4.1)
where we have 5et
A=-A[x~(0),0] = ~ e1n°A(")[x~(0)].
(4.2)
n=1
E4. (4.2) c011ect5 a11 A (n) 9au9e tran5f0rmat10n5 1nt0 a 51n91e 0ne 6y add1n9 an exp11c1t dependence 0n 0 0f the 5tr1n9 f1e1d A.
A. Ne0eu, P.C. We5t / 8050n1c5tr1n95
147
7he tran5f0rmat10n 0f the 5upp1ementary f1e1d5 4~(") under A~1) 15 1
~c6(n + 1) •.= "~n LnA(1)
(4.3)
8~ = a~r 2p 2A(1)
(4.4)
and we may rewr1te th15 a5 ,
where ,=,[x"(0),01=
12 e~°°--0(m+~)[x"(0)] • 2n
(4.5)
n:1
7here ex15t 51m11ar f0rmu1ae f0r X(n) and A(2). Hence, the 1nf1n1te 5et 0f A(n) 9au9e tran5f0rmat10n5 w111c011ect 1nt0 0ne w1th parameter A[x~(0), 0 ], and the 1nf1n1te 5et 0f 5upp1ementary f1e1d5 4,(,+1) w1116e c0nta1ned 1n ~[x"(0), 0]. 1t ha5 6een c1ear f0r 50me t1me that there mu5t ex15t 50me new n0n-a6e11an 1nvar1ance pr1nc1p1e that determ1ne5 the 5tructure 0f 1nteract10n5 1n 5tr1n9 the0r1e5. 7h15 1nvar1ance w111 h0pefu11y f0110w fr0m a new 9e0metr1c pr1nc1p1e and have a c0nceptua11y 51mp1e f0rm. We n0te that the pr0cedure 0ut11ned 1n the prev10u5 5ect10n 0n 5tr1n9 1nteract10n5 1n pr1nc1p1e determ1ne5 1t, and we remarked that the 5tructure c0n5tant5 are 1nt1mate1y re1ated t0 the matr1x e1ement5 0f the threere99e0n vertex. We a150 f1nd that the parameter 0f the 9au9e tran5f0rmat10n 15 0f the funct10na1 f0rm A[x~(0), 01. 1n the act10n c0n5tructed a60ve, there are ar61trar11y h19h 5p1n5 wh1ch are c0n515tent1y c0up1ed t09ether 1n a 9au9e- and L0rent2-1nvar1ant way. Wr1t1n9 th15 0ut 1n term5 0f c0mp0nent f1e1d5 w0u1d revea1 h0w the c0n515tency 0ccur5 and h0w the d1ff1cu1t1e5 enc0untered prev10u51y are 0verc0me. P055e551n9 the 1nteract1n9 act10n f0r a11 1eve15 w0u1d a110w f0r a 5tra19htf0rward 4uant12at10n 0f the the0ry. 0 n e 5e1ect5 a 9au9e and 1ntr0duce5 Faddeev-P0p0v 9h05t5 1n the n0rma1 way. At the hnear12ed 1eve1, 0ne w0u1d rec0ver the re5u1t5 0f ref. [4] ;n an appr0pr1ate 9au9e. Hav1n9 1ntr0duced 9au9e 1nvar1ance5 c0rre5p0nd1n9 t0 the tw0-d1men510na1 c0nf0rma1 9r0up, 0ne m19ht w0nder a60ut the1r an0ma11e5. F0r examp1e, 5tart1n9 fr0m 6050n1c 0pen 5tr1n95, 0ne 5h0u1d f1nd that c105ed 5tr1n95 are needed f0r c0n515tency at the 0ne-100p 1eve1. 7he 50(32) an0ma1y cance11at10n 0f ref. [18] mu5t 6e 0ne 0f an 1nf1n1te 5er1e5 0f cance11at10n5 ar151n9 fr0m the 9au91n9 0f the 5uperc0nf0rma1 tw0-d1men510na1 9r0up; the app11cat10n 0f the meth0d5 0f th15 paper t0 the c0var1ant f0rmu1at10n 0f 5uper5tr1n95 15 under 1nve5t19at10n [19]. Phen0men01091ca1 app11cat10n5 0f 5uper5tr1n95 re4u1re d1men510na1 reduct10n 6y 5p0ntane0u5 c0mpact1f1cat10n. Armed w1th c0var1ant 5tr1n9 e4uat10n5 0f m0t10n, 0ne c0u1d 100k f0r c1a551ca1 exact 501ut10n5 and exam1ne the1r pr0pert1e5.
148
A. Ne0eu, P.C. We5t / 8050n1c 5tr1n95
F0r 0ne 0f the auth0r5 (P.C.W.) th15 w0rk 6e9an w1th a 5er1e5 0f d15cu5510n5 w1th 5tuart Ra6y at L05 A1am05, where the w0rk 0f ref. [6] wa5 carr1ed 0ut. P.C.W. w0u1d 11ke t0 thank the L05 A1am05 1a60rat0ry f0r 1t5 h05p1ta11ty, and D. 51an5ky and 6 . We5t f0r d15cu5510n5.
Append1x A
We w111 n0w extend the re5u1t5 0f 5ect. 2 t0 the 51xth exc1ted 1eve1 0f the 0pen 5tr1n9. 7he 0n1y new A x and A 2 1nvar1ant c0m61nat10n5 apart fr0m 6 (6) at th15 1eve1 are ~,-(6) ~ 15d•(6) + 4L14~(5)• L46p(2)+ 12X(6),
~t(6) = 5dp(6) ~ 3L2d~(4)• 2L3~(3) + 16X(6) + 6L1X0).
(A.1) (A.2)
7 h e 0n1y term we can add t0 the act10n wh1ch 15 0n1y 61hnear 1n der1vat1ve5 and A 1 and A 2 1nvar1ant at th15 1eve1 15 •a ( f f ( 6 ) ~ ( 6 ) ) •1.. 6 (ff(6), fft(6)) ..}• 1 c (~ ~(6)ff/(6)) .
(A.3)
App1y1n9 the ~2~5) 5ymmetry 0f e4. (2.38), we f1nd that the act10n 15 1nvar1ant pr0v1ded that 96 + 4a = 2, 7he
86 + 1 8 c = 6.
(A.4)
9 (4) 5ymmetry d0e5 re4u1re 50me c0rrect10n5 term5, wh1ch 0n1y affect th15
1eve1, 6ef0re 1t can 6e a 5ymmetry. 7he c0rrect 12(4) tran5f0rmat10n5 are 84~~2)= - 2L• 2 ~ (4) "•]- ~L • 4L2~-~ (4) ,
8d~(3)~-~ 2 L ~ 1 ~ 2 0 ) ,
~(4) = • 49(4)
~ ( 5 ) = • 31~L~1L2~(4) ~ ~X (4) = 3 ~ (4) ,
~d,6(6) = • ~ L 2 ~ ( 4 ) ,
6X (6) = -- ~L2~2(4)
(A.5)
and they f0rm an 1nvar1ance pr0v1ded a -•~ ,x
6 - - 9 .1
(A.6)
7 h e re5u1t1n9 act10n p055e55e5 a further 5ymmetry wh1ch can 6e u5ed t0 9au9e away
A. Neveu, P.C. We5t / 8050n1c5tr1n95
149
X (6). 1t 15 0f the f0rm (~(2)
=
•
2L
• 4 ~ (6) -
2L•
8e:(4)= • 3L~1~2 4 ,,
8¢(6)~
~(6),
(~1~(3)= 2L• 1~2~,
2~ , ,
8e:(5)= 2 L ~ 1 ~ f 6), 8~((3) -- 3~2~,
~X (6) = 3Q(6),
(A.7)
where ~=
~3(L2~1-1-3L
2 ) ~ (6).
W e n0W 91Ve the C0mp1ete act10n at 51xth 1eVe1 Wh1Ch 15 1nVar1ant Under A(1) ... A (6) a n d ~2~2)... ~2(6): A = 1(++, (L0 - 1)4•) - 2(4 ~2), (L0 + 1 ) ~ {2))
- 1E
n=l
(a("), a
+ E
n=4
-1- 1~(~t(6), ~ ( 6 ) ) - 8(X(4), ( L 0 "+"3 + 1L~1L1)X(4) ) + 12( X ~4), L• 1X (5))
"•[-
9(X (5) , X ~5)).
(A.8)
0 n e m a y expect the tran5f0rmat10n5 0f e45. (A.5) and (A.7) t0 51mp11fy w1th the 1ntr0duct10n 0f further 9au9ed-away 5upp1ementary f1e1d5. 7 h e 9enera1 pattern 6e91n5 t0 emer9e at th15 1eve1 and w111 6e rep0rted e15ewhere.
Reference5 [1] Y. Nam6u, Pr0c. 1nt. C0nf. 0n 5ymmetr1e5 and 4uark m0de5, Detr01t 1969 (60rd0n and 8reach, NY, 1970) and C0penha9en 5ummer 1n5t1tute 1970 [2] P.A.M. D1rac, Lecture5 0n Quantum mechan1c5, (8e1fer 6raduate 5ch001 0f 5c1ence, Ye5h1va Un1ver51ty, New Y0rk, 1964); A. Han50n, 7. Re99e and C. 7e1te1601m, C0n5tra1ned ham11t0n1an 5y5tem5 (Acad. Na2. de1 L1nce1, R0me, 1976) [3] E. Cremmer and L-L. 6erva15, Nuc1. Phy5. 890 (1975) 410; M. Kaku and K. K1kkawa, Phy5. Rev. D10 (1974) 1110, 1823 [4] W. 51e9e1, Phy5. Lett. 1488 (1984) 556, 1498 (1984) 157, 162 [5] M.A. V1ra50r0, Phy5. Rev. D1 (1970) 2933 [6] R.C. 8r0wer, Phy5. Rev. D6 (1972) 1655; P. 60ddard and C.8. 7h0rn, Phy5. Lett. 408 (1972) 235 [7] R.C. 8r0wer and C.8. 7h0rn, Nuc1. Phy5. 831 (1971) 163 [8] 5. Ra6y and P.C. We5t, unpu6115hed
150
A. Neveu, P.C. We5t / 8050n1c 5tr1n95
[9] 7. 8ank5 and M. Pe5k1n, Pr0c. 5ymp. 0n An0ma11e5, 9e0metry and t0p0109y, Ar90nne Nat. La6. March 1985; M. Kaku and J. Lykken, 161d [10] D. Fr1edan, Un1v. 0f Ch1ca90 prepr1nt EF1 85-27 (Apr11 1985) [11] P. van N1euwenhu12en, Nuc1. Phy5. 860 (1973) 478 [12] L.P.5. 51n9h and C.R. Ha9en, Phy5. Rev. D9 (1974) 898 [13] H. van Dam and M. Ve1tman, Nuc1. Phy5. 822 (1970) 397 [14] L. Cane5ch1, A. 5chw1mmer and 6. Vene21an0, Phy5. Lett. 308 (1969) 351 [15] R.C. 8r0wer and J.H. We15, Lett. Nu0v0 C1m. 3 (1970) 285 [16] 5. 5c1ut0, Lett. Nu0v0 C1m. v01. 11 9 (1969) 411 [17] J. 5cherk, Nuc1. Phy5. 831 (1971) 222 [18] M.8.6reen and J.H. 5chwar2, Phy5. Lett. 1498 (1984) 117 [19] A. Neveu, H. N1c01a1 and P. We5t, 1n preparat10n
6 A U 6 E C0VAR1AN7 L 0 C A L F 0 R M U L A 7 1 0 N 0 F 8050N1C 57R1N65 A. NEVEU and P.C. WE57* CERN,
6eneva,
5w1t2er1and
Rece1ved 1 Ju1y 1985
A free 6050n1c 5tr1n9 f1e1d act10n 15 c0n5tructed that 15 10ca1 and 1nvar1ant under the 1nf1n1te 5et 0f 9au9e tran5f0rmat10n5 c0rre5p0nd1n9 t0 the c0nf0rma1 9r0up 0f the tw0-d1men510na1 w0r1d 5heet. 7h15 a110w5 f0r a 5tra19htf0rward 9au9e c0var1ant meth0d 0f 5ec0nd-4uant121n9 5tr1n9 f1e1d the0ry. 7he meth0d 0f exten510n t0 the 1nteract1n9 ca5e 15 pre5ented.
1. 1ntr0duet10n 7 h e c1a551ca1 6050n1c 5tr1n9 15 de5cr16ed 6y the act10n [1] A
1 2~ra~ f d 0 d * [ d e t
0~x"0~x~] 1/2,
wh1ch 15 1nvar1ant under tw0-d1men510na1 reparametr12at10n 0f the w0r1d 5heet. F0r the 0pen 5tr1n9, x~(a, , ) 15 def1ned f0r 0 ~< a ~<~r w1th 60undary c0nd1t10n5 00x ~ = 0 at a = 0, ~r. 1t 15 c0nven1ent t0 mathemat1ca11y extend th15 ran9e fr0m - ,r < a ~<~r 6 y 1ett1n9 x ~ ( - a, ~-) = x~(a, ,). 7he c105ed 5tr1n9 15 def1ned 6y x~(0, ,), -~r ~< a < ~r, per10d1c 1n a w1th per10d 2~r. A5 a re5u1t 0f th15 reparametr12at10n 1nvar1ance 1t 15 a c0n5tra1ned 5y5tem. 1n the c1a551ca1 ham11t0n1an f0rmu1at10n 0f the 0pen 5tr1n9, the c0n5tra1nt5 may 6e wr1tten 1n the f0rm
L.=~ra~
f,,
(
1 0x~)2,
d ° e - 1 " ° P~-~ 21ra~ 00
(1.1)
where
8A • ~(a) •
p ~, = -
-
0x~ ar
k ~, = •
7 h e L.•5 06ey the a19e6ra
[Ln, Lm]=(n--171)Ln+m, * Permanent addre55:K1n9•5 C011e9e, L0nd0n WC2, UK. 125
(1.2)
126
A. Neveu, P.C. We5t / 8050n1c 5tr1n95
wh1ch 15 ju5t that 0f the 1nf1n1te-d1men510na1 c0nf0rma1 9r0up 0f the tw0-d1men510na1 w0r1d 5heet. 70 f1r5t 4uant12e the 5y5tem we ad0pt the u5ua1 c0rre5p0ndence 6etween P01550n 6racket5 and c0mmutat10n re1at10n5 and take [x~(0),x~(a~)] = 0,
[P~(0), P~(0~)] = 0,
[x"(0), P~(0~)] = 1h ,/~t~(0 - 0•).
(1.3)
7he re1at10n5 are repre5ented 6y the 0perat0r5
x"(0) = x " ( 0 ) ,
6 P"(0) = -1h3x~(0)
.
(1.4)
7he 5tate 0f the 5tr1n9 15 de5cr16ed at a 91ven ~- 6y a funct10na1 0f x"(0), +[x"(0)]. A1th0u9h the ham11t0n1an van15he5, 0ne may u5e the 4uant12at10n meth0d5 0f c0n5tra1nt5 d15cu55ed 6y D1rac [2]. 1n th15 paper, we 5ha11 91ve a meth0d 0f 5ec0nd 4uant121n9 the 5tr1n9 that 15 free fr0m c0n5tra1nt5 and p055e55e5 9au9e 1nvar1ance5 c0rre5p0nd1n9 t0 the 1nf1n1te d1men510na1 c0nf0rma1 9r0up. F0r the 0pen 5tr1n9, 0ne 0f the 1nf1n1te 5et 0f 9au9e tran5f0rmat10n5 w1116e Yan9-M1115 tran5f0rmat10n5, wh11e f0r the c105ed 5tr1n9 they w111c0nta1n 9enera1 c00rd1nate tran5f0rmat10n5. 1n th15 way, the 9au9e 1nvar1ance 0f the 5tr1n9 c0nta1n5 a11 prev10u51y kn0wn 9au9e 5ymmetr1e5, and 1n the 9au9e c0var1ant act10n, 0ne f1nd5, at the 11near12ed 1eve1, the Yan9-M1115 and E1n5te1n act10n5. Up unt11 recent1y, the 5ec0nd 4uant12at10n 0f 5tr1n95 ha5 e1ther 6een perf0rmed w1th c0n5tra1nt5 6e1n9 pre5ent 0r carr1ed 0ut 1n a 91ven 9au9e 5uch a5 the 119ht-c0ne 9au9e [3], where the c0n5tra1nt5 have 1n effect 6een 501ved. Quant12at10n u51n9 8R5 techn14ue5 0f the 11near12ed the0ry 1n a 91ven 9au9e ha5 6een d15cu55ed 1n ref. [4]. 1t 15 p055161e that 0ne can, 1n pr1nc1p1e, d15c0ver a11 the pr0pert1e5 0f a the0ry 6y 4uant121n9 1n a 91ven 9au9e. H0wever, w1th0ut the a6111ty t0 u5e a11 the w15d0m ac4u1red w1th 5ec0nd 4uant12at10n, th15 may 6e d1ff1cu1t.An examp1e 15 the c0mputat10n 0f an0ma11e5 wh1ch are a65ent 1n the 119ht-c0ne 9au9e; h0wever, a carefu1 5earch 0f L0rent2 1nvar1ance w0u1d 5h0w that 1t 15 v101ated when 9au9e an0ma11e5 are pre5ent. A150, the wh01e 5u6ject 0f n0n-pertur6at1ve 5em1-c1a551ca1 phen0mena and 5p0ntane0u5 5ymmetry 6reak1n9 ha5 up t0 n0w 0n1y 6een deve10ped 1n the 9au9e 1nvar1ant framew0rk. 1t 15 a150 p055161e that the f1n1tene55 pr0pert1e5 0f 5tr1n95 may 6ec0me part1cu1ar1y apparent 1n a c0var1ant f0rmu1at10n a5 they d1d 1n the ca5e 0f 5uper5ymmetr1c the0r1e5. 70 5ec0nd 4uant12e the 5y5tem we 5uppre55 a5 u5ua1 the • dependence 1n the wave funct10na1 ~p[x~(0)] and re4u1re an act10n 5 wh1ch 15 u5ed t0 we19h the Feynman path 1nte9ra1. 7he vacuum-t0-vacuum amp11tude 15 91ven 9enera11y 6y
f
exph 5,
(1.5)
A. Neveu, P.C. We5t / 8050n1c 5tr1n95
127
where the act10n 15 0f the 9ener1c f0rm
5 = f ~ x , ( 0 ) f(4J [ x , ( 0 ) ] ) + 9au9e f1x1n9 + 9h05t + 50urce term5.
(1.6)
1n a c0var1ant f0rmu1at10n, 5 w1116e 1nvar1ant under 9au9e 1nvar1ance5 c0rre5p0nd1n9 t0 the tw0-d1men510na1 c0nf0rma1 9r0up. We w111pre5ent 5uch an act10n f0r the 11near12ed the0ry. F0r a 11near12ed the0ry, the 9au9e 1nvar1ance5 are a6e11an. We pre5ent the 5trate9y f0r 6u11d1n9 5tr1n9 1nteract10n5 5tart1n9 fr0m the 11near12ed the0ry. 7h15 15 perf0rmed 6y a type 0f N0ether techn14ue, and 0ne f1nd5 that the A6e11an and r191d 1nvar1ance5 0f the r191d the0ry 6ec0me kn1tted t09ether. 7he kn1tt1n9 1nv01ve5 the three-re99e0n c0up11n9. 7he dependence 0f x ~ 0n 0 1n the 0pen 5tr1n9 can 6e wr1tten 1n the f0rm
x~(0)=
~, n=
e1~°x,, "
(1.7)
--00
where x~(0) 15 def1ned f0r -~r < 0 < ~r and x~ ~ = x~. 7he tw0-d1men510na1 c0nf0rma1 9r0up are 9enerated 6y the V1ra50r0 0perat0r5 L, wh1ch are def1ned 6y ,~,
f
L, = 1~ra~
d 0 e - ~ " ° 6 " ( 0 ) 6 ~ ( 0 ) ~/,,,
(1.8)
where /6~(°)=-1h8x~(0)
1 0x ~ F2~ra~ 00
(1.9)
1t 15 we11 kn0wn [5] that the 0n-5he11 5tate5 0f the D = 26 6050n1c 0pen 5tr1n9 are 91ven 6y (L0 - 1)~k = 0, Ln~6 = 0
(1.10) f0r n > 0.
(1.11)
1t ha5 6een 5h0wn [6] that th15 5y5tem p055e55e5 0ne tachy0n at 1t5 10we5t ma55 1eve1, a11 0ther 5tate5 hav1n9 2er0 0r p051t1ve ma55e5 and that a11 5tate5 are 9h05t free, 1.e., have pr0pa9at0r5 w1th p051t1ve re51due5. A c0rre5p0nd1n9 c0var1ant pr0ject0r wa5 f1r5t wr1tten d0wn 1n ref. [7]. 70 ana1y2e e45. (1.10) and (1.11), 1t 15 m0re c0nven1ent t0 1ntr0duce creat10n and de5truct10n 0perat0r5 def1ned 6y the e4uat10n 1
+00
/%(0) = ~.(2a,)1/2 ,,=E~ 0 / n, e 1,,,.
(1.12)
128
A. Neveu, P.C. We5t / 8050n1c 5tr1n95
U51n9 e4. (1.7) we f1nd that 0
Fr0m the rea11ty 0f P~(0) we f1nd that a ~ n = a~t. 7he a~5 c0mmutat10n re1at10n5 are
[a.",
:0,
[a~, a~,t ] = n3n, m3""
(1.14)
f0r n, m >1 0. 7he V1ra50r0 0perat0r5 [5] can 6e expre55ed 1n term5 0f the a,•5 6y +~
L.
E
~-
""
01m0Ln
-
m
(1.15)
where L 0 15 under5t00d t0 6e n0rma1 0rdered. 7hey 06ey the m0d1f1ed a19e6ra
[ L,, Lm1= ( n - rn)L,+ m + ~n(n 2 - 1 ) 3., m.
(1.16)
We f1nd 1n part1cu1ar that
L 0=
+ ~ a,,a,,"~t ,
1-~vP~22~0
(1.17)
m=1
L 1 = L* 1 =
" "1 -~0~0a
~ -"*,~ ~m~m+1
(1.18)
m=1
where
(0ff) 1/2 0 a~ = -1 -~
0x"
We may wr1te the 5tate 0f the 5tr1n9 1n 0CCupat10n num6er 6a515 1n term5 0f the creat10n 0perat0r5 a~* 6y +[x"(0)]=
(~0(x)+1A~a~t+h,,a~*a~*+;a"tA2+, 2 , ~}(x~(0)10).
(1.20)
7he vacuum 5at15f1e5 the e4uat10n a~1 (x~(0)10) = 0,
n>~1.
(1.21)
A1ternat1ve1y, 0ne c0u1d chan9e 6a515 t0 d1a90na112e x"(0), and th15 ha5 a1ready 6een c0n51dered 1n the 119ht-c0ne f0rmu1at10n 0f 1nteract1n9 5tr1n95 [3]. 7he vacuum
A. Neveu,P.C. We5t / 8050n1c5tr1n95
129
0f e4. (1.21) 15 0f the f0rm
(1.22)
7he act10n 0f the a,~* 0n (x~(0)10> pr0duce5 the we11-kn0wn c0mp1ete 5et 0f Herm1te p01yn0m1a15. 1n term5 0f c0mp0nent f1e1d5 we f1nd that e4. (1.10) ha5 a5 c0n5e4uence5
(1)
=
[ (1) 0 2--~
h~,. . . . . etc...
(1.23)
7 h e act10n 0f e4. (1.11) 15 0~A~ = 0 = V~1 0~h~, + A~2) = v~ 0~A~2) + h•.
(1.24)
We n0w w15h t0 f1nd an act10n wh1ch d0e5 n0t have the c0n5tra1nt5 0f• e4. (1.11) 6ut 1n5tead p055e55e5 9au9e 1nvar1ance5 that a110w the c0n5tra1nt5 0f e4. (1.11) t0 ar15e a5 a 9au9e ch01ce. C0n5e4uent1y, we expect an 1nf1n1te num6er 0f 9au9e 1nvar1ance5, 0ne f0r every 0ne 0f the c0n5tra1nt5 wh1ch 9enerate the c0nf0rma1 9r0up. 7h15 can 6e ach1eved ma55 1eve1 6y ma55 1eve1 6y 5ucce551ve1y re1ea51n9 the c0n5tra1nt5 0n ~k. At the f1r5t 1eve1, we re1ea5e the c0n5tra1nt L~k = 0 6ut ~p 15 5t1115u6ject t0 L 2 ~ = L~4~ = L 3 ~ = L2L1~k... = 0.
(1.25)
C0n51der n0w the 9au9e tran5f0rmat10n 0f 4J
9ap = L 1 A 1 ,
(1.26)
L 1 A 1 = L 2 A 1 = 0.
(1.27)
where A 1 15 5u6ject t0
U51n9 the f0rm 0f L• 1 91ven 1n e4. (1.18), we f1nd that th15 1nvar1ance c0nta1n5 the tran5f0rmat10n 8A1 = 0~,A1(x ) wh1ch 15 the a6e11an tran5f0rmat10n expected f0r a 11near12ed Yan9-M1115 the0ry. An act10n 1nvar1ant under 8~ = L•1A 115 91ven 6y ~ / ~ ( L 0 2 ~ -- 1 - 1 ~ L ~ 1 L 1 ) ~ p ) .
(1.28)
7he e4uat10n 0f m0t10n 15 91ven 6y (L 0-1-~L
1L1)~k=0.
(1.29)
130
A. Neve~ P.C. We5t / 8050n1c 5tr1n95
Exp11c1t1y te5t1n9 th15, we f1nd that ( L 0 - 1 - •L•1L1)L•1A
1 = (L0L• 1 - L 1 - L-1L0)A1 = 0,
(1.30)
51nce L1A 1 = 0. E4. (1.29) wa5 pr06a61y kn0wn t0 a few pe0p1e 1n the 01d heyday 0f 5tr1n9 the0ry, 6ut ha5 6een red15c0vered m0re recent1y [8, 9]. 7he pr0ject0r P 0f a 5tr1n9 f1e1d 0nt0 the phy51ca1 5tate5 0f e45. (1.10) and (1.11) wa5 91ven 1n ref. [7]. At 10we5t 0rder 1t 15 91ven 6y 1 P = 1 - ~ L ~ 1 - ~ 0 L 1. (1.31) We n0te that at th15 1eve1 the e4uat10n 0f m0t10n 15 91ven 6y ( L 0 - 1)P~6 = 0.
(1.32)
1t ha5 6een pr0p05ed recent1y [9,10] that e4. (1.32) 15 the c0rrect e4uat10n 0f m0t10n 0f a11 1eve15 0f the 5tr1n9, and th15 5pecu1at10n ha5 6een enc0ura9ed 6y the a60ve c01nc1dence f0r the 5p1n 0ne at the f1r5t 1eve1. 1n ref5. [9] and [10], P ha5 6een f0rma11y c0mputed f0r a11 1eve15, and e4. (1.32) 15 exp11c1t1y n0n10ca1 at the 5ec0nd 1eve1 and m0re and m0re 50 f0r h19her 1eve15.0ne•5 5u5p1c10n5 are further ar0u5ed 6y the fact that P can 6e c0n5tructed 1n an ar61trary 5pace-t1me d1men510n, and that D = 26 15 n0t part1cu1ar1y fav0ured. 7he c1eare5t way t0 5h0w that e4. (1.32) 15 n0t the r19ht 9enera112at10n 15 t0 c0n51der the f1r5t 1eve1 0f the c105ed 6050n1c 0r1ented 5tr1n9. At th15 1eve1, the c0var1ant de9ree5 0f freed0m are de5cr16ed 6y a 51n91e 5ymmetr1c f1e1d h,,. 7he V1ra50r0 c0nd1t10n 1mp11e5 that 1t 5at15f1e5 02h~ = 0,
0~h~ = 0.
(1.33)
7he5e e4uat10n5 te11 u5 that at th15 1eve1 the c105ed 5tr1n9 c0nta1n5 0n1y a 5p1n tw0 and a 5p1n 2er0. 7he 9enera112at10n 0f e4. (1.32) f0r th15 1eve1 15 02RfRxh0x=0, where
(1.34)
Rf= (~f 0.00 --
02
)"
(1.35)
7he reader 1mmed1ate1y rec09n12e5 that th15 15 a n0n-10ca1 e4uat10n, wh1ch d0e5 n0t adm1t a ham11t0n1an f0rmu1at10n. Mak1n9 1t 10ca1 6y mu1t1p11cat10n 6y 0 2 1ead5 t0 add1t10na1 5tate5. 1n fact, 1t 15 kn0wn [11] that there 15 n0 L0rent2 1nvar1ant, 9au9e 1nvar1ant, 10ca1 act10n c0n5tructed fr0m h,, a10ne, wh1ch de5cr16e5 60th 5p1n tw0 and 5p1n 2er0. 7he
131
A. Ne0eu, P.C. We5t / 8050n1c5tr1n95
0n1y Way t0 aCh1eVe th15 15 t0 1ntr0dUCe an0ther 5Ca1ar f1e1d t0 de5Cr16e the 5p1n 2er0; th15 15 the We11-kn0wn E1n5te1n + ma551e55 5Ca1ar act10n. A5 We 5ha11 5ee, rather than the 5p1n 0ne, the f1r5t 1eVe10f the C105ed 5tr1n9 111U5trate5 the 9enera1 pattern, name1y, f0r a 10Ca1 f0rmU1at10n, the the0ry natura11y re4U1re5 5Upp1ementary f1e1d5, a5 We 5ha11 n0W dem0n5trate.
2. 6au9e c0var1ant f0rmu1at10n 0f the 11near12ed 5tr1n9 f1e1d the0ry 2.1. 0PEN 57R1N65 A5 0Ur 5tart1n9 p01nt, We take the act10n at the f1r5t 1eVe1, e4. (28) ~( ~ , ( L 0 - - 1 - - ~ L - 1 L 1 ) ~ 6
(2.1)
),
wh1ch 15 1nvar1ant under the 9au9e tran5f0rmat10n5 ~1~6=L~1A1,
(2.2)
8 2 ~ = L ~ 2 A 2,
where A1, A 2 and ~k are 5u6ject t0 L 1 A , = L 2 A 1 = 0,
L2~ = L2~ = L3~ = L 2 L 1 ~ . . . . .
0.
(2.3)
We 5ha11 n0w f1nd the c0var1ant f0rmu1at10n 1eve1 6y 1eve1, 5y5temat1ca11y re1ax1n9 c0nd1t10n5 (2.3). At the 5ec0nd 1eve1 1n 26 d1men510n5, the 0n-5he11 5tate5 are 0ne ma551ve pure ••5p1n tw0•• [ - a n 50(25) 5ymmetr1c trace1e55 ten50r]. 7he the0ry 0f wave e4uat10n5 f0r part1c1e5 0f a 91ven 5p1n 15 a c0n5e4uence 0f the the0ry 0f 1nduced repre5entat10n5 0f the P01ncar6 9r0up. 1t 15 a 5y5temat1c pr0cedure t0 f1nd the pr0ject10n 0perat0r5 0nt0 the phy51ca1 5tate5 0f a 91ven 5p1n. 1n the 5tudy 0f the5e 0perat0r5 0ne f1nd5 the we11-kn0wn wave e4uat10n5. 1n fact the part1c1e5 0f 5p1n 2er0, • and 1 are the except10n rather than the ru1e, 1n that mu1t1p1y1n9 the1r pr0ject10n 0perat0r5 6y 8 2 y1e1d5 a 10ca1 f1e1d e4uat10n. F0r 5p1n5 9reater than 0ne, mu1t1p11cat10n 6y 8 2 1ead5 t0 a n0n-10ca1 and 50 an unaccepta61e f1e1d e4uat10n. 7he pr0cedure t0 6e f0110wed 1n the5e ca5e5 wa5 w0rked 0ut 1n ref. [12] and 1t 1nv01ve5 the 1ntr0duct10n 0f 5upp1ementary f1e1d5. F0r examp1e ••pure 5p1n tw0•• 15 de5cr16ed 0n-5he11 6y the f1e1d h ~ = h ~ 5u6ject t0 the e4uat10n5
8~h~= h~ = ( 82-- m2)h~v=0.
(2.4)
7he pr0ject0r 15 we11 kn0wn [13]. 1t 1nv01ve5 term5 0f the f0rm 0p 0X0p0a//(82) 2 and hence mU1t1p11Cat10n (8 2) d0e5 n0t 1ead t0 a 10Ca1 f1e1d e4Uat10n. 1n fact, there 15 n0 Way t0 de5Cr16e 1n a L0rent2 C0Var1ant Way 0n1y a ma551Ve 5p1n tW0 part1C1e 1n term5 0f 0n1y h~v = hv~, 5U6jeCt t0 h~" = 0. 7he C0rreCt e4Uat10n5 0f m0t10n 1nV01Ve the
132
A. Neveu, P.C. We5t / 8050n1c 5tr1n95
1ntr0duct10n 0f a 5upp1ementary f1e1d ~ and are 91ven 6y
( - 0 2 + m2)h~, + ( 0~0°h0, + 0,00h0~ ) - 2 ~ 0 ° 0 x h 0 x D-2(
6~,
- D - 1
0 r 0 v~ -- ~ -
0~0"h~=
) 0 2~
,
0 2 - D -D 2m2) d~,
(2.5)
where D 15 the d1men510n 0f 5pace-t1me. 1ndeed, the5e c0up1ed e4uat10n5 1ead t0 the de51red re5u1t, name1y
0 = 3~h~,~= 4~= (0 2 - m2)h~,~.
(2.6)
7he 5tr1n9 f1e1d ~k[x"(0)] c0nta1n5 at the 5ec0nd 1eve1 h,, and A~2), [5ee e4. (1.20)]; h0wever, A (2) a5 we 5ha11 5ee, 15 9au9ed away and the a60ve d15cu5510n 1mp11e5 that we re4u1re 0ne extra 5upp1ementary f1e1d 4~t0 1mp1ement the 5p1n tw0 e4uat10n. 7h15 f1e1d 4~ w111 6e the 10we5t c0mp0nent 0f a 5upp1ementary 5tr1n9 f1e1d 4~(2)[x~(0)]. We n0w re1ea5e 50me 0f the c0n5tra1nt5 (2.3) 0f the f1r5t 1eve1, and ~k and 4~(2)are n0w 5u6ject t0 =
L1~(2) = L 2 ~ (2) . . . . .
. . . . .
0,
0.
(2.7)
7he m05t 9enera1 expre5510n 0f the c0rrect 0rder 15
(L 0 - 1 -1L~1L1-1~yL~2L2)1~-~- (L21--{-- 3/3L~ 2)t~(2) = 0,
(2.8)
( L~ + ~/3L2) 4J= ( aL 0 + 6)~ ~2).
(2.9)
7he u5e 0f the 5ame 13 1n (2.7) and (2.8) 15 re4u1red 6y demand1n9 that the e4uat10n5 0f m0t10n f0110w fr0m an act10n. An a1ternat1ve way 0f 5earch1n9 f0r a 9au9e 1nvar1ance 15 t0 demand that L 1 0n e4. (2.8) 5h0u1d van15h when we u5e e4. (2.9). Carry1n9 th15 0ut and u51n9 the c0n5tra1nt5 0f e4. (2.7), we f1nd
~L~1( 1 1 L 2 + 3yL2)~ + (4L~1L0 + 2L~1)67(2) + 9/3L~1d~(2)=0.
(2.10)
70 e11m1nate ~k 6y e4. (2.9) re4u1re5 y =/3 and we f1nd
•L•1(aL 0 + 6)ep <2)+ L•1(4L 0 + 2 + 9/3),p<2)= 0.
(2.11)
133
A. Neveu, P.C. We5t / 8050n1c 5tr1n95
Hence we c0nc1ude that a -- 8 and 6 = 4 + 9/3. App1y1n9 L 2 1n a 51m11ar way f1xe5 /3 = 1, and we f1nd the e4uat10n5 0f m0t10n
L0•
-
• n=1 ~ L • ,9/L , ) ,/, + (L 1 + 3 L
0,
( L 2 + 3L2) tp = (8L 0 + 13)4~(2) .
(2.12)
7h15 5y5tem 0f e4uat10n5 15 1n fact 1nvar1ant under the 9au9e tran5f0rmat10n5
~ p = L~xA1,
~1(~(2) =
~$2~6 = L • 2 A 2 ,
1LaA1,
(~2~(2)= 3 A 2 ,
(2.13)
w1th
L2A 1 = L2A1 . . . . .
0,
L1A 2 "~- L2A 2 .
0.
. . . .
(2.14)
We 5tre55 that e45. (2.12) are A 2 1nvar1ant 0n1y f0r D = 26. 1t may 6e p055161e, w1th the 1ntr0duct10n 0f further 5upp1ementary f1e1d5, t0 re1ax th15 c0nd1t10n. 7he c0rre5p0nd1n9 act10n 15 91ven 6y
•( tp, ( L 0 - 1 - ~ L ~ 1 L 1 - 1 L ~ 2L2)~p ) +( ~,( L2~1 + 3L~ 2)4,(2)) -~(ep(2),(8L0 + 13)e~(2)).
(2.15)
7h15 c0mp1ete5 the ana1y515 0f the 5ec0nd 1eve1, and we n0w 90 t0 the th1rd and f0urth 1eve15. At the th1rd 1eve1, we have a5 0n-5he11 5tate5, 0ne ma551ve vect0r, 0ne ant15ymmetr1c 5ec0nd rank ten50r and 0ne • pure 5p1n-three••. 7he 1atter 15 1n effect de5cr16ed 6y the f1e1d ~ p = 4~(~p) 5u6ject t0 8~4~ 0 = 0 = 4~p. 7h15 re4u1re5 the 5upp1ementary f1e1d5 4~ and 4~ t0 de5cr16e 1t5 pr0Pa9at10n. 7he e~ f1e1d 15 5upp11ed at th15 1eve1 6y the 5ec0nd c0mp0nent f1e1d 0f e~(2), 1.e., 4~(2)[x~(0)] -- [4~(2)(x) + a.~*a, .~ ~ , + • • • ] (x~(0)10).
(2.16)
Hence, we 0n1y re4u1re 0ne new 5upp1ementary 5tr1n9 f1e1d 4,(3)[x~(0)] 51nce the ant15ymmetr1c ten50r and the vect0r d0 n0t re4u1re 5upp1ementary f1e1d5. 0 n e can carry 0ut the c0unt1n9 0f 5tate5 at the f0urth 1eve1 and 0ne f1nd5 that 0ne new 5upp1ementary 5tr1n9 f1e1d 15 re4u1red ~(4)[x"(0)].
134
A. Net)e•u, P.C. We5t / 8050n1c 5tr1n95
At the f0urth 1eve1 the f1e1d5 are 5u6ject t0
0 = L54J = L ~ = L4L1~6... etc.,
0 = L3~6(2)= L2Laep~2)= L3ep~2)... etc., 0 = L246(3)= L2~6(3)... etC., 0 = L14~4)... etc.
(2.17)
7 h e act10n 15 06ta1ned a5 f0r the 5ec0nd 1eve1; 0ne wr1te5 d0wn a11 p055161e term5 0f the c0rrect 0rder and re4u1re5 the re5u1t t0 6e 9au9e 1nvar1ant. 7he act10n 15 f0und t0 6e
((
21 ~
41
L0--1--E-2-~L-nLn n=1
4-(~,(L
~d
"4- (~,(L214-3L~2)4~ (2))
2L•14- 8L 3)4~(3)) 4• ( ~ , ( L ~ 3 L
14- ~ L ~ 4)~)(4))
---12(4~(2),(8L 04- 134- 2L•1L 1 - - L 2L2)~ (2)) 1(~6(3), (8L0 4• ~)4~(3)) -- 1(4~(4), (12L ° + 2~7)4~(4)) -- 3( L2* (2), ,(4, ) -- 6( L14~(2), ~(3)) -- 2( L2* (2), L14~(3)) -- 10( L1~ (3), 4~(4)). (2.18) 1t 15 1nvar1ant under the 9au9e tran5f0rmat10n5 1
311p = L•1A1,
81¢~(n+1) = ~ n L . A 1 ,
32~k=L 2A2,
32ep~2)=3A2+~L~2L2A 2,
1 1 324~0) = ~L1A 2 - 9L• xL2A 2,
62d•(4) = ~L2A 2 .
(2.19)
Further 9au9e tran5f0rmat10n5 can 6e f0und 6y tak1n9 5pec1a1 va1ue5 0f A 1 and A 2, 5ay A 1 = + L 2A 3,A 2= -L•1A3 wh1ch 91ve5 8~6= L 3A 3 and expre5510n5 f0r 3ff~"). 51nce the 11near12ed act10n (2.18) 15 4uadrat1c 1n the 5upp1ementary f1e1d5, 0ne c0u1d 1n pr1nc1p1e 1nte9rate them 0ut, rec0ver1n9, up t0 that 1eve1, the pr0ject0r 0f ref. [7]. 1n 0rder t0 d0 th15, h0wever, 0ne mu5t 1nvert the matr1x 0f the 4uadrat1c f0rm 1n the 5upp1ementary f1e1d5, a pr0ce55 wh1ch 15 a1ready very pa1nfu1 at f0urth 1eve1.
135
A. Neveu, P.C. We5t / 8050n1c5tr1n95
8ef0re c0nt1nu1n9 t0 the f1fth 1eve1, 1t 15 u5efu1 t0 ref0rmu1ate the re5u1t at the f0urth 1eve1 6y 1ntr0duc1n9 a further 5upp1ementary f1e1d. 7h15 a150 a110w5 u5 t0 91ve 50me 5ect0r5 0f the act10n t0 a11 1eve15. 7he ~k e4uat10n 0f m0t10n 15 0f the f0rm
L 0 - 1 - )-•. ~ n L~ , L , + + 5upp1ementary f1e1d term5 = 0.
(2.20)
n=1 7h15 f0rm 15 nece55ary 1n 0rder t0 e11m1nate the LmL 0 term5 up0n the mu1t1p11cat10n 6y L,.. Act1n9 w1th L 1 we f1nd the re5u1t
1
.+2
]
n=1 ~ n L - n L1Ln "•[•- "-n-~1~1Ln +1 1~ -•[- 5Upp1ementary f1e1d term5
0. (2.21)
1n 0rder t0 e11m1nate the + term we re4u1re 5upp1ementary f1e1d5 ~(n+1) wh05e e4uat10n5 0f m0t10n are n+2 ) L1L,, + n n - - - ~ L , + 1 + = 5upp1ementary f1e1d5.
(2.22)
F0r the act10n 0f L2 0n e4. (2.20), we f1nd the re5u1t 3 ~( L12 + ~L2)~6~
8 L n (L2Ln+n ~L~1(L1L2+~L3)~ ~ __~_n
n=2
n+ n+
+ 5upp1ementary f1e1d term5 = 0.
(2.23)
We 065erve that the f1r5t tw0 term5 1nv01ve the 5ame 0perat0r5 a5 0ccur f0r the app11cat10n 0f Lp 1n fact, 0ne can ehm1nate 4~ fr0m th15 e4uat10n u51n9 the 1dent1ty
n(n+4) L2 L n -1
n+2
2 2
(
Ln+2=~Ln(L1+ 3L2)-2L 1 L1Ln+n
2 n2-2n-2[ 3 n+1
n+2 ) Ln+1 n+1
(n+1)(n+3) ] L1L"+1 -~
n+2
Ln+ 2 . (2.24)
H0wever, 1t 15 c0nven1ent t0 1ntr0duce a further 5et 0f 5upp1ementary f1e1d5 X(") (n >1 4), the e4uat10n 0f m0t10n f0r Xt") 6e1n9 n2-4 L2Ln - 2 +
n
) L. ~6= 5upp1ementary f1e1d term5.
(2.25)
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A. Neveu, P.C. We5t / 8050n1c 5tr1n95
7he5e e4uat10n5 f0110w fr0m the act10n
• ~p, L 0 - 1 -
, L . ~p +
,=1 2n
~, L ,L 1 + n -
n=1
n+ 1
L n-1 -
n(n+4,L ) ) n+ 2
n-2 X(•+2)
(~(n+1)
(2.26)
p1u5 5upp1ementary term5 a10ne. Vary1n9 + we f1nd the ~p e4uat10n t0 6e 0f the f0rm
L0-1-
+
n~1
-~nL . L . ~,+
n=1
L ,L 1 + n -
n + 1
n
L ~ , L ~ 2 + n - ~ - - 2 L ~ , ~ 2 X(~+2)= 0.
1
(2.27)
Fr0m th15 e4uat10n we may read 0ff the A 1 and A 2 9au9e tran5f0rmat10n5 t0 a11 0rder5. 5u65t1tut1n9 ~p = L ~ A ~ , we f1nd 1
~1~ =
L1A1,
81~(n+1) =
"~nL.A1,
3~X(n) = 0.
(2.28)
Wh11e 5u65t1tut1n9 3~6 = L 2A 2, we f1nd ~2t~ (2) "~- 3A 2,
32~0) = •L1A 2, 1
~2(~ ( n ) = 0,
f0r n >1 4,
~X (n+2) =
~nLnA2 .
(2.29)
We c0u1d further 51mpf1fy the a60ve f0rmu1ae 6y 1ntr0duc1n9 tw0 5upp1ementary f1e1d5 f0r each 0f the expre5510n5 ( L 2 + 3(L2))~ and (L1L 2 + 8(L3))+. 7h15 c0rre5p0nd5 t0 X (2), X (3) f0r the act10n 0f L 2 and ~(2) and (/)(3) f0r the act10n 0f L 1. 7he act10n, h0wever, w111 0n1y c0nta1n the c0m61nat10n5 0 (2) + X (2) and 4~(3)+X(3). 7h15 fact c0u1d 6e enf0rced 6y demand1n9 that the act10n have the 5ymmetr1e5 ~(2) •••) ~(2) •4• ~(2),
X (2) ~ X (2) • ~ (2),
4~(3)~ ~(3) + ,4(3),
X0) ~ X(3) • f/(3).
(2.30)
A. Ne0e~ P.C. We5t / 8050n1c 5tr1n95
137
1n th15 ca5e the re5u1t1n9 9au9e tran5f0rmat10n5 are 1
~1~(n+ 1)= .~n LnA1 ~
814~= L•1A1, ~xX~") = 0, 814 = L• 2A 2,
32~ (n) = 0 ,
1
~2X~"+2) = ~n L, A 2,
n>~1, (2.31)
82X ~2)= ~A 2.
Hencef0rth we w111re9ard X(2) and X(3) a5 6e1n9 a65ent 0r 9au9ed away 6y ~(2) and f20). 1t 15 1n5truct1ve t0 c0mp1ete the 54uare5 1n the act10n 0f e4. (2.26). 1t 6ec0me5 0~
~(4~, ( L 0 - 1)4~) - • ~
1
- - ( 6 ~ " ) , 6~")) + 5upp1ementary f1e1d5 term5, (2.32)
n ~ 1 21•/
where 6 ( n ) = Lnt~ -- 2 ( n 2 -- 1) dp(n) -- 2 n L - 1 ~ ( n + 1 ) -
2 r t L - 2 X (n+2) -- 2 ( n 2 -- 4 ) X ( n ) .
(2.33)
Under A 1 a11 the 6 (") are 1nert except 6 ~1), wh11e under A 2 they tran5f0rm a5 826(1) =
L2L1A2,
826(3) = --
~26 ~") = 0
~26(2)= 4 ( L 0 + 1)A 2 - 2 L 1 L 1 A 2 ,
3L1A 1, f0r n >1 4.
(2.34)
An0ther u5efu1 c0m61nat10n 15 ~-~,+3) = n (n + 2)4~("+3) + (n + 1)L1~ ~"+2) - L , + 145~1) + 3(n + 1)X(n+3) . (2.35) 7h15 15 a150 1nert under A 1 and tran5f0rm5 under A 2 a5
~2~~4)= L~A 2,
32~") = 0,
n >1-5.
(2.36)
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•4. Neveu, P.C. We5t / 8050n1c5tr1n95
1t 15 n0W a 51mp1e matter t0 Ver1fy the 1nVar1anCe 0f the f0110w1n9 act10n at f0Urth 1eVe1 Under the A (1) and A (2) 9aU9e 1nVar1anCe5 4
1(~p,(L 0 - 1 ) ~ p ) - ~ E
1
~---n(6("), 6("))
n=1
- 2(~6(2), (L0 + 1)d,~(2)) -1-1(~(4), ~(4)) -- 8(X(4), ( L 0 4- 3 ) X ( 4 ) ) .
(2.37)
7he reader may w0nder what the re1at10n 15 6etween the a60ve f0urth 1eve1 act10n and that 91ven prev10u51y 1n e4. (2.18). 1n fact, X(4) can 6e 9au9ed away 51nce the a60ve act10n (2.37) p055e55e5 the add1t10na1 5ymmetry 34~(2) = - 2L • 2 ~ (4) , 3~(4) = -- ~ ( 4 )
8~ (3) = 2L• a12(4), ~(4) = 382(4).
(2.38)
Up0n 9au91n9 away X (4) w e 06ta1n the prev10u5 act10n 0f e4. (2.18) and the prev10u5 tran5f0rmat10n5 1aw5 0f e4. (2.19) pr0v1ded we make the nece55ary c0mpen5at1n9 tran5f0rmat10n5. Hav1n9 6u11t up the a60ve f0rma115m, the f1fth 1eve1 15 ea511y f0und. 7he act10n 0f e4. (2.37) 15 A 1 and A 2 1nvar1ant pr0v1ded we add the term5 12 ( X (4), L•1, X(5)) + 9(X (5), X (5)) + 4(L1X (4), L1X(4)). 7he 0n1yA 1 and A 2 1nVar1ant term we may add t0 the act10n 0f e4. (2.37) 15 c (~.(5), ~.(5)),
(2.39)
a5 we11 a5 extend1n9 the 5um 0f the 6n~5 fr0m 4 t0 5 . 7 h e c0eff1c1ent c 0f th15 new term 15 ea511y f0und 6y demand1n9 that the ~2(4) 5ymmetry 1nher1t5 t0 the f1fth 1eve1. 0 n e f1nd5 that c = + 2. 7 h e f1fth 1eve1 then ha5 a new 5ymmetry wh1ch 15 91ven 6y 3~ (2) = - 2L• 312(5), 8 ~ (4) = 2 L • 1~ (5),
3X (4) = 0,
3~ (3) = 0, 34~(5)= •f2 (5),
3X (5) = 3~2(5) ,
(2.40)
wh1ch a110w5 0ne t0 9au9e away X (5). 7h15 5ymmetry /2 (5) and that 0f ~(4) a r e an 1nvar1ance 0f the ~p e4uat10n 0f m0t10n due t0 the 1dent1ty (2.24). 7he 51xth 1eve1 15 pre5ented 1n append1x A. We w111 n0w dem0n5trate that we can rec0ver e45. (1.10) and (1.11) 6y a 5er1e5 0f 9au9e ch01ce5. 0 f c0ur5e, 51nce 0ne ha5 an act10n 1nvar1ant under ar61trary 9au9e tran5f0rmat10n5, we mu5t ch005e a very 5pec1f1c 9au9e 1n 0rder t0 arr1ve at e45.
139
A. Neveu, P.C. We5t / 8050n1c 5tr1n95
(1.10) and (1.11). F1r5t1y, We 9aU9e X(") = 0 U51n9 12("); We then U5e A (2), A (3), A(4) ... t0 9aU9e away 4~(2),4~0), ~(4) . . . . Up0n mak1n9 the nece55ary c0mpen5at1n9 tran5f0rmat10n5, the re5u1t1n9 e4uat10n5 w1th 4,(2) = ff0) = 4,(4) = 0 are 5t111 1nvar1ant under A 0). We may u5e A (t) t0 5et Lt~p = 0. 5u65t1tut1n9 Lt~p = d~(2) = d~(3) = 4,(4) . . . . . 0, we rec0ver e45. (1.10) and (1.11). 7h15 dem0n5trate5 that the act10n c0n5tructed a60ve d0e5 1ndeed de5cr16e the c0rrect 5pectrum f0r the 0pen 6050n1c 5tr1n9. 7h15 pr0cedure 15 50mewhat 5u6t11e and make5 u5e 0f 0n-5he11 9au9e 1nvar1ance5 1n a 51m11ar way t0 the way 1n wh1ch 0ne dem0n5trate5 that QED de5cr16e5 a ph0t0n w1th 2 de9ree5 0f freed0m. 7he reader may ver1fy, f0r h1m5e1f the a60ve 5tatement. 2.2. CL05ED 57R1N65 7 h e c105ed 5tr1n9 x ~ ( 0 ) can 6e wr1tten 1n the f0rm +00 E e1n°Xn9
X9( 0 ) =
~t ~
where x " = x "
(2.41)
--00
We def1ne the creat10n and de5truct10n 0perat0r5 6y 3
P.(0)
1
t 3x~,
-
=
1
0x ~
--=
1~Xt~ 2rra• 00
+00
~ ~., •/r(2a•) 1/2 n=~-00 -1n, t*
2~ra~ 0 0 1
P~(0)
0x ~
---+----=
=
1
+ ~0
..
rr(2a•) 1/2 n=~00 e-1~"a"
(2.42)
7 h e V1ra50r0 0perat0r5 are 91ven 6y L.=
• 1 r a ~f ]
d0e-1"°P~
( )02
+00
= ~1 m ~
£ , = ~ra~
f~
d0e/"°
- - •R
~ ~(
0) 2 =
~, m
..
E
a~a,~
m,
- - ~
+00
-~-~
09 m 0~ n ~
--
-- rn
"
(2.43)
c10
7 h e a•5 5at15fy the re1at10n5
[~2,~*] =n~.,m~
[ ~:., ~*] = .an, m~ , [am,a,]
f0r n , m > ~ 1 ,
=0
(2.44)
1t 15 we11 kn0wn that the 0n-5he11 5tate5 0f the 6050n1c 0r1ented c105ed 5tr1n9 are de5cr16ed 6y the e4uat10n5 ( L 0 + 2 0 - 2 ) + = 0, ( L 0 - L0)4* = 0,
L,~6 = L,,~k = 0,
n >1 1.
(2.45)
140
A. Neveu, P.C. We5t / 8050n1c 5tr1n95
At the f1r5t 1eve1, the5e e4uat10n5 have a ma551e55 ••pure 5p1n 2•• and 0ne 5p1n 2er0. A5 d15cu55ed 1n the 1ntr0duct10n, the5e 5tate5 can 0n1y 6e de5cr16ed 1n a 10ca1, 9au9eand L0rent2-c0var1ant way 6y the f1e1d5 h,~ and 4~. 51nce the 5tr1n9 f1e1d ~ 0n1y c0nta1n5 h ~ at the f1r5t 1eve1, we mu5t add a 5upp1ementary 5tr1n9 f1e1d ~(1). 7he f1r5t 1eve1 re5u1t 15 f0und 1n much the 5ame way a5 f0r the 0pen 5tr1n9; we 6e91n 6y re1ax1n9 50me 0f the c0n5tra1nt5 0n 4~, wh1ch 15 n0w 5u6ject t0 (2.46)
(L0 - L0)~ = L~k = L2~ = 212~k= L2~P= 0 a5 we11 a5
L1~6(1)= /7-,1~(1)=0. 7he act10n 15 f0und t0 6e ~(4~, ( L 0 + L0 - 2 - L
,L 1
L~1L1)~) "~-(1p,L
-
1L~1~ (1))
~(~(1) (L0..1~20)~(1))
(2.47)
and 15 1nvar1ant under the tran5f0rmat10n5
~1~k= L-1A1 4- L-1A1,
~1 dp(1) = L1A1 + L1A1•
(2.48)
m
where the 9au9e parameter5 A 1 and A~ are 5u6ject t0 ( L 0 - L 0 + 1)A1= 0,
( L 0 - L 0 - 1)A1= 0,
L1A 1=0, L2A 1 = L1A 1 =
L,A 1=0, 0,
L2A 1 = L1A 1 =
0.
(2.49)
7he act10n 0f e4. (2.47) at the f1r5t ma55 1eve1 can 6e eva1uated 1n term5 0f the c0mp0nent f1e1d5 h ~ and 4~, and 15 n0ne 0ther, m0du10 a tr1v1a1 f1e1d redef1n1t10n, than the 11near12ed E1n5te1n p1u5 5ca1ar act10n. 7he treatment 0f h19her 1eve15pr0ceed5 1n e55ent1a11y the 5ame way a5 f0r the 0pen 5tr1n9. We treat here 0n1y the next (5ec0nd) 1eve1, where the • 5p1n•• can 6e a5 h19h a5 f0ur. 0 n e f1r5t arran9e5 the A1, A1 9au9e 1nvar1ance5, re1ax1n9 appr0pr1ate1y the c0nd1t10n5 (2.49). 0ne then c0n5truct5 an act10n 1nvar1ant under Ap A~, 1ntr0duc1n9 the nece55ary 5upp1ementary f1e1d5. After that, 0ne 1mp05e5 the new A 2, -42 1nvar1ance5. 7he c0nd1t10n5 (2.49) are n0w rep1aced 6y 0 = L2a1 = L
51 = E 2 A 1 = E1 A1 = L15
= E A2 = c 7,2 =
(2.50)
Up0n app1y1n9 th15 pr0cedure, 0ne f1nd5 that f1ve new 5upp1ementary f1e1d5
141
A. Neveu, P.C. We5t / 8050n1c 5tr1n95 -
•
(02, 02, 02, X 2, X 2) are needed at the 5ec0nd 1eve1, w1th tran5f0rmat10n 1aw5 3102 = L1A1,
~1~2 = L1A1,
~1X2 ~---L 2 A 1 ,
~10~2 = 0,
(~202
: 3A 2•
L2A1,
820=L~2A2+2~2A2,
~2~2 = 3A 2,
~2X 2 = 2 L 1 A 2•
~1X2 =
~2X2 = 2L1A2, (2.51)
~20~2= L2A 2 + L2A 2"
At th15 1eve1, the 1nvar1ant act10n 15 ~(4~, ( L 0 + E 0 - 2
- L•1L
1
1 - L•1L
1 - •L
~(01,(L0~{~-J7-~0)01)~-~(~,L
+(~,(2~-1+~2
•
2L2 - 1J~,~ 2L2) 1~) --}-(1/J, L•1L 1
101)
t
2L 2 0 ~ ) - 7 ( 0 2 , ( L 0 + L 0 + 2 ) 0 ~ ) •
2)~2)+~(~,L 12 212)+~(~,2 1L 2x2)
-(01, L 12102)- (01, 2 1L1~2)- ( ~ , ( L0 + ~0 + ~)~2) 1 -1 + ~(0~, L~1L1~2) ~r~,~,2 L 112)- (02,( L0 + L0 + ~)0~) + ~(02, L 1L10~) 5 1 L - - 2 ( 0 2 , 1-~-1X2)--4(X2,( 0~-1~-~04-1)X2) ~(X2, L - 1 L 1 f ( 2 ) - •(X2, (L0 + L0 + 1)X2) 1 - ~(x2,
L~1L1x~)
-
3 • ~(0~, L202 + L2~2) + ~ ( ~ L ~ 1
+1(X2L~1-02L~2,21X2-L202)-~(0~2,(L0+
- 0 2 2 ~ 2, L122 - L~02) L 0 + 2)0~2).
(2.52)
3. 1nteract10n5
1n th15 5ect10n, we w111dem0n5trate h0w t0 c0n5truct the 1nteract10n5 f0r the 0pen 6050n1c 5tr1n9 and carry 0ut the f1r5t 5tep 0f th15 pr0cedure. 70 f1nd the 1nteract1n9 the0ry fr0m 1t5 11near12ed f0rm, we w111 emp10y a 9enera112at10n 0f the N0ether meth0d. 1n th15 meth0d, 0ne 5tart5 w1th the 11near12ed the0ry wh1ch p055e55e5 a6e11an 10ca1 1nvar1ance5 and a 51m11arnum6er 0f r191d 1nvar1ance5. 0 n e then make5 the r191d 1nvar1ance5 10ca1 and f1nd5 an 1nvar1ant n0n-11near act10n 0rder 6y 0rder 1n the c0up11n9 c0n5tant. 7h15 15 ach1eved 6y kn1tt1n9 t09ether the r191d and 10ca1 a6e11an 1nvar1ance5. F0r the 11near12ed the0ry 91ven 1n the prev10u5 5ect10n, the 11near12ed a6e11an 1nvar1ance5 are the tran5f0rmat10n5 w1th parameter5 A(1), A(2), A(3), etc. 7he r191d 1nvar1ance5 are the u5ua1 r191d r0tat10n5 0f the f1e1d5 under the 9r0up 6 . 1n fact,
142
A. Neveu, P.C. We5t / 8050n1e 5tr1n95
there are an 1nf1n1te 5et 0f 5uch 1nvar1ance5, 51nce we can make an 1ndependent r0tat10n at every ma55 1eve1. 7he f1r5t 5uch 1nvar1ance 15 the 0ne wh1ch r0tate5 a11 ma55 1eve15 1n the 5ame way; 1t 15 91ven 6y
~ = [~-~(1),~],
(3.1)
x~(0). Let u5 den0te the nth r191d r0tat10n 6y t2(n). t2
(3.2) where 7"1• and 72 are 0perat0r5 0f the f0rm
=
(3.3)
c)100)105)10c)
wh1ch are t0 6e determ1ned. 7he 1a6e15 a, 6 and c c0rre5p0nd t0 the three ••1e95•• 0f 7~ wh1ch are 111u5trated 1n f19. 1: 7here a150 ex15t 51m11ar e4uat10n5 f0r the 5upp1ementary f1e1d5 [4~(~)). 1n th15 5en5e, the meth0d we are u51n9 repre5ent5 a 9enera112at10n 0f the u5ua1 N0ether meth0d. Up0n 5u65t1tut1n9 the n0w-10ca1 var1at10n5 0f ]~p) 0f e4. (3.2) and th05e f0r ]4~(n)) 1nt0 the 11near12ed act10n, we w11106ta1n an expre5510n wh1ch 15 11near 1n (t2 (n) ] and 6111near 1n [4J) and J~(")). 7he act10n 0f the L,•5 wh1ch make up the 11near12ed act10n d0e5 n0t, 1n 9enera1, 06ey the L1e6n1t2 ru1e. Hence the 7•5 w111 have t0 6e very 5pec1a1 0perat0r5, 50 that the act10n 0f the L,•5 pr0duce5 further L,•5 0n the 0ther 1e95. 1n the d15cu5510n wh1ch f0110w5, we w111 0n1y refer t0 t2(1), 6ut the exten510n t0 t2 (~) 15 1n pr1nc1p1e c1ear. 7he var1at10n 0f the 11near12ed act10n can then 6e wr1tten 1n the f0rm
(~2(x) 1L.8 ~) ,
61~ua>=a
72•
1Q6 >
1~c> F19. 1.
+ a~
(3.4)
14J6> 1Qc>
A. Neveu, P.C. We5t / 8050n1c 5tr1n95
143
where 5 (•) 15 an expre5510n 6111near 1n 14~) and 14)(•)). 7h15 var1at10n 0f the 11near12ed act10n can 6e cance11ed 6y the add1t10n 0f a term 0f 0rder 9 t0 the act10n. 7he var1at10n 0f th15 term at 0rder 90 re5u1t5 fr0m 0n1y the var1at10n 0f the f1e1d5 under the a6e11an 1nvar1ance5 wh05e parameter A(1) 15 1dent1f1ed 6y 1
1A(~)> = -1~2(x)). 9
(3.5)
($20) 1L15(1)
(3.6)
F0r examp1e, a term 0f the f0rm
15 cance11ed 6y the add1t10n 0f the term 0f the f0rm 9{4J15 (1) .
(3.7)
7he var1at10n5 0f th15 new act10n at 0rder 9 are then cance11ed 6y add1n9 new term5 t0 the act10n and p055161y new term5 t0 the tran5f0rmat10n 1aw5. 7h15 15 repeated unt11 0ne 06ta1n5 an act10n 1nvar1ant t0 a11 0rder5 1n 9. Fr0m the 119ht-c0ne f0rmu1at10n 0f the 1nteract1n9 5tr1n9 f1e1d the0ry and the 9e0metr1ca1 1nterpretat10n 0f 1nteract10n5 5uch a5 5p11tt1n9, j01n1n9 and rearran9ement 0f 5tr1n95, 0ne can expect at m05t 4uart1c 1nteract10n5 f0r 0pen 5tr1n95 and cu61c 0ne5 f0r c105ed 5tr1n95 a150 1n a 9au9e c0var1ant f0rmu1at10n. H0wever, the vert1ce5 w111 nece55ar11y 1nv01ve the 5upp1ementary f1e1d5 4~(~). 7he 0ccurrence 0f 1nteract10n5 at 1ea5t 4uadrat1c 1n ~(~) re5u1t5 1n the pract1ca1 1na6111ty t0 1nte9rate 0ut the 5upp1ementary f1e1d51n the path 1nte9ra1. 5uch term5 are 1nev1ta61e; f0r examp1e, there ex15t5 1n the c105ed 5tr1n9 5ect0r the u5ua1 9rav1t0n-ma551e55 5ca1ar c0up11n9, even 6ef0re tak1n9 the f1e1d the0ry 11m1t, hence a n0n-2er0 d~(1)~ (1) 1nteract10n term. 7h15 5h0w5 that the n0n-10ca1 f0rmu1at10n 0f the free 5tr1n9 the0ry 0f ref5. [9] and [10] can def1n1te1y n0t 6e 9enera112ed t0 the 1nteract1n9 ca5e. 7he a60ve pr0cedure can 6e carr1ed 0ut 1eve1 6y 1eve1, and we w111 n0w f1nd the 0rder 9 1nteract10n5 at the f1r5t 1eve1. We 6e91n w1th the act10n
1-•L
(3.8)
U51n9 the var1at10n5 0f 14J) 0f e4. (3.2), we f1nd (4•61(L0 - 1 - ~L-~L1)6{(4,a1(~2c171 + <~2.1 (~c1 72~ } •
(3.9)
7h15 mu5t 6e expre55161e 1n the f0rm ($2a1L~(4%1 (4%1W,
(3.10)
144
A. Ne0eu, P.C. We5t / 8050n1c 5tr1n95
wh1ch we can cance1 w1th the var1at10n 0f the term 99( ~1(~k6[ (~P¢1W.
(3.11)
C0mpar1n9 e45. (3.9) and (3.10), we re4u1re the 0perat0r5 7, and W t0 5at15fy
L~W= (L 0 - 1 -
1L•1L1)671
-~- ( L 0 - 1 - 1 L ~ 1 L 1 ) c 7 2 .
(3.12)
L0n9 a90, a cyc11ca11y 5ymmetr1c three-p01nt vertex wa5 wr1tten d0wn [14] 1n the f0rm
(-1)mY(n)
V = exp { - n=1 (01a-n016-m-]- 016-n0Lc m~1- 01t~-n0£a-m) 1~(m ~ - 1 ~ F ~
}
2~1 -}- 1) 10) "
m=0
(3.13) 1t 15 1n5truct1ve t0 eva1uate V, tak1n9 the externa1 5tate5 t0 c0nta1n 0n1y the ma551e55 Yan9-M1115 vect0r5; 0ne f1nd5 the re5u1t k1" E3e1" e2 +
k2"
E1E2"E3 -1- k 3 " e2e1" 83 "4- k1
" E3k2 " e1k3" E2-
(3.14)
7he 1a5t term, 0f c0ur5e, d15appear5 1n the 2er0 510pe 11m1t. 7h15 vertex d1ffer5 fr0m 0ne wh1ch 15 ant15ymmetr1c under the 1nterchan9e 0f any tw0 externa1 11ne5 6y 10n91tud1na1 term5 wh1ch are: E~E~[~k~p
+ 1~
10
,~1,~3,~2
+
1~
~ 0
(3.15) We theref0re take 0ur 0perat0r5 t0 6e 0f the f0rm
W ~ [1-Jr0t(La-1 + L6-1 + Lc1) +f1( L6-1Lc-1 + L~-1L6-1 d- U -1 L ~-1] 4- 8L ~ 1L6 1 L c 1 ]
V,
71= Y(1+ a•Lc1)V, (3.16)
72 = E(1 "~-0L~( L6 1 + Lc 1 + La-1)) V"
7he5e are the m05t 9enera1 0perat0r5, tak1n9 1nt0 acc0unt the c0n5tra1nt5 f0r the f1r5t 1eve1 wr1tten 1n the prev10u5 5ect10n: L 11~)
=
L21~k) = L~6p) . . . . .
0.
(3.17)
A. Neveu, P.C. We5t / 8050n1c5tr1n95
145
( L ~ - L~ + L6~1- L6 + L ~ ) V = 0 ,
(3.18)
U51n9 the re5u1t [15]
and the ana1090u5 re5u1t5 06ta1ned 6y cyc11ca11ypermut1n9 a, 6 and c, we can eva1uate L~ 0n W and f1x the a60ve c0eff1c1ent5. 0ne f1nd5 W=[1
--
1
a
L 6
2(L-1+-1
+ Lc•1)
1[ La--1 L 6- 1 + L 6--1 L c--1 + L c- 1 L ~- 1 ]~- - ~L ~ 1L 6--1 L ~--1J] V , 2
-•[- 2~
71=V, 72= --(1--(L6•1+La1+Lc•1))V
,
(3.19)
where 7~(6, a, c) = 71(c, 6, a ) .
(3.20)
At th15 1eve1, 72• happen5 t0 6e cyc11ca11y5ymmetr1c, and the permutat10n (3.20) 15 1rre1evant. 1t may 6ec0me re1evant 1n next 0rder. Hence the 0rder 9 re5u1t at the f1r5t 1eve115 ~(#1L 0 - 1 -
~9<~k~1<#61 <~kc1W.
(3.21)
11a26> + <61
(3.22)
~L1L 16k5 -
7h15 act10n 15 1nvar1ant t0 0rder 90 under 1
~1~6>
=-L6 9
-
1n the 11terature 0f f1fteen year5 a90 0n dua1 m0de15, there appeared 0ther three-Re99e0n vert1ce5. 0ne, 1,V, 15 ju5t W w1th the 0pp051te cyc11c 0rder1n9 0f externa1 11ne5. 80th W and W were 06ta1ned fr0m the 5c1ut0 [16] vertex 5 6y app1y1n9 appr0pr1ate tw15t 0perat0r5 12t~5t, acc0rd1n9 t0 f19. 2 and ref5. [14] and [15]. 1t ha5 6een 5h0wn 1n ref. [17] that 0ne 06ta1n5 1n the 2er0-510pe 11m1t the c0rrect Feynman d1a9ram5 0f a f1e1d the0ry 6y 5umm1n9 0ver tw15ted and untw15ted 1nterna1 11ne5 0f the 5tr1n9 d1a9ram. 7h15 re5u1t 5h0w5 that 0ne 5h0u1d 1nc1ude a11 f0ur vert1ce5 W, 1~, 5, 5 0f f19. 2 (5u1ta61y m0d1f1ed 6y Ln~5 1n the manner 5h0wn a60ve f0r W) 1n the act10n and tran5f0rmat10n 1aw5. 7he re1at1ve c0eff1c1ent5 mu5t 6e determ1ned 6y c105ure 0f the a19e6ra 0f 9au9e tran5f0rmat10n5. Fr0m (3.22) and (3.19), 1t 15 06v10u5 that the 5tructure c0n5tant5 0f th15 a19e6ra are e55ent1a11y 91ven 6y the matr1x e1ement5 0f the three-re99e0n vert1ce5. 7h15 w1116e dea1t w1th 1n a future pu611cat10n.
146
A. Neveu, P.C. We5t / 8050n1c 5tr1n95
//
~
U
= a
~C 6
N
Q~w15t 5 ~t"
~c
F19. 2. 4. 0ut100k 50me n0ve1 4ua11tat1ve feature5 emer9e fr0m the ana1y515 0f the prev10u5 5ect10n wh1ch we expect t0 carry t0 a11 0rder5 1n 5p1te 0f techn1ca1 c0mp11cat10n5. 7he5e feature5 are 6e5t 5een and exp1a1ned 6y 901n9 t0 funct10na1 f0rm. 1n 5uch a f0rm, under A ~"), the 9au9e tran5f0rmat10n 0f ~k may 6e wr1tten a5
f 47 d0e1"0p2(°) n=1 -~
=~a~p2(0)Ad0,
(4.1)
where we have 5et
A=-A[x~(0),0] = ~ e1n°A(")[x~(0)].
(4.2)
n=1
E4. (4.2) c011ect5 a11 A (n) 9au9e tran5f0rmat10n5 1nt0 a 51n91e 0ne 6y add1n9 an exp11c1t dependence 0n 0 0f the 5tr1n9 f1e1d A.
A. Ne0eu, P.C. We5t / 8050n1c5tr1n95
147
7he tran5f0rmat10n 0f the 5upp1ementary f1e1d5 4~(") under A~1) 15 1
~c6(n + 1) •.= "~n LnA(1)
(4.3)
8~ = a~r 2p 2A(1)
(4.4)
and we may rewr1te th15 a5 ,
where ,=,[x"(0),01=
12 e~°°--0(m+~)[x"(0)] • 2n
(4.5)
n:1
7here ex15t 51m11ar f0rmu1ae f0r X(n) and A(2). Hence, the 1nf1n1te 5et 0f A(n) 9au9e tran5f0rmat10n5 w111c011ect 1nt0 0ne w1th parameter A[x~(0), 0 ], and the 1nf1n1te 5et 0f 5upp1ementary f1e1d5 4,(,+1) w1116e c0nta1ned 1n ~[x"(0), 0]. 1t ha5 6een c1ear f0r 50me t1me that there mu5t ex15t 50me new n0n-a6e11an 1nvar1ance pr1nc1p1e that determ1ne5 the 5tructure 0f 1nteract10n5 1n 5tr1n9 the0r1e5. 7h15 1nvar1ance w111 h0pefu11y f0110w fr0m a new 9e0metr1c pr1nc1p1e and have a c0nceptua11y 51mp1e f0rm. We n0te that the pr0cedure 0ut11ned 1n the prev10u5 5ect10n 0n 5tr1n9 1nteract10n5 1n pr1nc1p1e determ1ne5 1t, and we remarked that the 5tructure c0n5tant5 are 1nt1mate1y re1ated t0 the matr1x e1ement5 0f the threere99e0n vertex. We a150 f1nd that the parameter 0f the 9au9e tran5f0rmat10n 15 0f the funct10na1 f0rm A[x~(0), 01. 1n the act10n c0n5tructed a60ve, there are ar61trar11y h19h 5p1n5 wh1ch are c0n515tent1y c0up1ed t09ether 1n a 9au9e- and L0rent2-1nvar1ant way. Wr1t1n9 th15 0ut 1n term5 0f c0mp0nent f1e1d5 w0u1d revea1 h0w the c0n515tency 0ccur5 and h0w the d1ff1cu1t1e5 enc0untered prev10u51y are 0verc0me. P055e551n9 the 1nteract1n9 act10n f0r a11 1eve15 w0u1d a110w f0r a 5tra19htf0rward 4uant12at10n 0f the the0ry. 0 n e 5e1ect5 a 9au9e and 1ntr0duce5 Faddeev-P0p0v 9h05t5 1n the n0rma1 way. At the hnear12ed 1eve1, 0ne w0u1d rec0ver the re5u1t5 0f ref. [4] ;n an appr0pr1ate 9au9e. Hav1n9 1ntr0duced 9au9e 1nvar1ance5 c0rre5p0nd1n9 t0 the tw0-d1men510na1 c0nf0rma1 9r0up, 0ne m19ht w0nder a60ut the1r an0ma11e5. F0r examp1e, 5tart1n9 fr0m 6050n1c 0pen 5tr1n95, 0ne 5h0u1d f1nd that c105ed 5tr1n95 are needed f0r c0n515tency at the 0ne-100p 1eve1. 7he 50(32) an0ma1y cance11at10n 0f ref. [18] mu5t 6e 0ne 0f an 1nf1n1te 5er1e5 0f cance11at10n5 ar151n9 fr0m the 9au91n9 0f the 5uperc0nf0rma1 tw0-d1men510na1 9r0up; the app11cat10n 0f the meth0d5 0f th15 paper t0 the c0var1ant f0rmu1at10n 0f 5uper5tr1n95 15 under 1nve5t19at10n [19]. Phen0men01091ca1 app11cat10n5 0f 5uper5tr1n95 re4u1re d1men510na1 reduct10n 6y 5p0ntane0u5 c0mpact1f1cat10n. Armed w1th c0var1ant 5tr1n9 e4uat10n5 0f m0t10n, 0ne c0u1d 100k f0r c1a551ca1 exact 501ut10n5 and exam1ne the1r pr0pert1e5.
148
A. Ne0eu, P.C. We5t / 8050n1c 5tr1n95
F0r 0ne 0f the auth0r5 (P.C.W.) th15 w0rk 6e9an w1th a 5er1e5 0f d15cu5510n5 w1th 5tuart Ra6y at L05 A1am05, where the w0rk 0f ref. [6] wa5 carr1ed 0ut. P.C.W. w0u1d 11ke t0 thank the L05 A1am05 1a60rat0ry f0r 1t5 h05p1ta11ty, and D. 51an5ky and 6 . We5t f0r d15cu5510n5.
Append1x A
We w111 n0w extend the re5u1t5 0f 5ect. 2 t0 the 51xth exc1ted 1eve1 0f the 0pen 5tr1n9. 7he 0n1y new A x and A 2 1nvar1ant c0m61nat10n5 apart fr0m 6 (6) at th15 1eve1 are ~,-(6) ~ 15d•(6) + 4L14~(5)• L46p(2)+ 12X(6),
~t(6) = 5dp(6) ~ 3L2d~(4)• 2L3~(3) + 16X(6) + 6L1X0).
(A.1) (A.2)
7 h e 0n1y term we can add t0 the act10n wh1ch 15 0n1y 61hnear 1n der1vat1ve5 and A 1 and A 2 1nvar1ant at th15 1eve1 15 •a ( f f ( 6 ) ~ ( 6 ) ) •1.. 6 (ff(6), fft(6)) ..}• 1 c (~ ~(6)ff/(6)) .
(A.3)
App1y1n9 the ~2~5) 5ymmetry 0f e4. (2.38), we f1nd that the act10n 15 1nvar1ant pr0v1ded that 96 + 4a = 2, 7he
86 + 1 8 c = 6.
(A.4)
9 (4) 5ymmetry d0e5 re4u1re 50me c0rrect10n5 term5, wh1ch 0n1y affect th15
1eve1, 6ef0re 1t can 6e a 5ymmetry. 7he c0rrect 12(4) tran5f0rmat10n5 are 84~~2)= - 2L• 2 ~ (4) "•]- ~L • 4L2~-~ (4) ,
8d~(3)~-~ 2 L ~ 1 ~ 2 0 ) ,
~(4) = • 49(4)
~ ( 5 ) = • 31~L~1L2~(4) ~ ~X (4) = 3 ~ (4) ,
~d,6(6) = • ~ L 2 ~ ( 4 ) ,
6X (6) = -- ~L2~2(4)
(A.5)
and they f0rm an 1nvar1ance pr0v1ded a -•~ ,x
6 - - 9 .1
(A.6)
7 h e re5u1t1n9 act10n p055e55e5 a further 5ymmetry wh1ch can 6e u5ed t0 9au9e away
A. Neveu, P.C. We5t / 8050n1c5tr1n95
149
X (6). 1t 15 0f the f0rm (~(2)
=
•
2L
• 4 ~ (6) -
2L•
8e:(4)= • 3L~1~2 4 ,,
8¢(6)~
~(6),
(~1~(3)= 2L• 1~2~,
2~ , ,
8e:(5)= 2 L ~ 1 ~ f 6), 8~((3) -- 3~2~,
~X (6) = 3Q(6),
(A.7)
where ~=
~3(L2~1-1-3L
2 ) ~ (6).
W e n0W 91Ve the C0mp1ete act10n at 51xth 1eVe1 Wh1Ch 15 1nVar1ant Under A(1) ... A (6) a n d ~2~2)... ~2(6): A = 1(++, (L0 - 1)4•) - 2(4 ~2), (L0 + 1 ) ~ {2))
- 1E
n=l
(a("), a
+ E
n=4
-1- 1~(~t(6), ~ ( 6 ) ) - 8(X(4), ( L 0 "+"3 + 1L~1L1)X(4) ) + 12( X ~4), L• 1X (5))
"•[-
9(X (5) , X ~5)).
(A.8)
0 n e m a y expect the tran5f0rmat10n5 0f e45. (A.5) and (A.7) t0 51mp11fy w1th the 1ntr0duct10n 0f further 9au9ed-away 5upp1ementary f1e1d5. 7 h e 9enera1 pattern 6e91n5 t0 emer9e at th15 1eve1 and w111 6e rep0rted e15ewhere.
Reference5 [1] Y. Nam6u, Pr0c. 1nt. C0nf. 0n 5ymmetr1e5 and 4uark m0de5, Detr01t 1969 (60rd0n and 8reach, NY, 1970) and C0penha9en 5ummer 1n5t1tute 1970 [2] P.A.M. D1rac, Lecture5 0n Quantum mechan1c5, (8e1fer 6raduate 5ch001 0f 5c1ence, Ye5h1va Un1ver51ty, New Y0rk, 1964); A. Han50n, 7. Re99e and C. 7e1te1601m, C0n5tra1ned ham11t0n1an 5y5tem5 (Acad. Na2. de1 L1nce1, R0me, 1976) [3] E. Cremmer and L-L. 6erva15, Nuc1. Phy5. 890 (1975) 410; M. Kaku and K. K1kkawa, Phy5. Rev. D10 (1974) 1110, 1823 [4] W. 51e9e1, Phy5. Lett. 1488 (1984) 556, 1498 (1984) 157, 162 [5] M.A. V1ra50r0, Phy5. Rev. D1 (1970) 2933 [6] R.C. 8r0wer, Phy5. Rev. D6 (1972) 1655; P. 60ddard and C.8. 7h0rn, Phy5. Lett. 408 (1972) 235 [7] R.C. 8r0wer and C.8. 7h0rn, Nuc1. Phy5. 831 (1971) 163 [8] 5. Ra6y and P.C. We5t, unpu6115hed
150
A. Neveu, P.C. We5t / 8050n1c 5tr1n95
[9] 7. 8ank5 and M. Pe5k1n, Pr0c. 5ymp. 0n An0ma11e5, 9e0metry and t0p0109y, Ar90nne Nat. La6. March 1985; M. Kaku and J. Lykken, 161d [10] D. Fr1edan, Un1v. 0f Ch1ca90 prepr1nt EF1 85-27 (Apr11 1985) [11] P. van N1euwenhu12en, Nuc1. Phy5. 860 (1973) 478 [12] L.P.5. 51n9h and C.R. Ha9en, Phy5. Rev. D9 (1974) 898 [13] H. van Dam and M. Ve1tman, Nuc1. Phy5. 822 (1970) 397 [14] L. Cane5ch1, A. 5chw1mmer and 6. Vene21an0, Phy5. Lett. 308 (1969) 351 [15] R.C. 8r0wer and J.H. We15, Lett. Nu0v0 C1m. 3 (1970) 285 [16] 5. 5c1ut0, Lett. Nu0v0 C1m. v01. 11 9 (1969) 411 [17] J. 5cherk, Nuc1. Phy5. 831 (1971) 222 [18] M.8.6reen and J.H. 5chwar2, Phy5. Lett. 1498 (1984) 117 [19] A. Neveu, H. N1c01a1 and P. We5t, 1n preparat10n