I L NUOVO CIMENTO VoL. 30 A, N. 1 1 Novembre 1975

A Canonical and Lorentz-Covariant Quantization of Yang-Mills Theories (*).

G. C u ~ c I

Istituto di Fisica dell'Universit~ - Pisa

R. FERRARI (**)

International Centre ]or Theoretical Physics - Trieste

(ricevuto il 20 Fcbbraio 1975; manoscrit to revisionato ricevuto il 2 Agosto 1975)

Summary . - - A Lorentz-covariant quantizat ion of non-Abelian gauge theories is performed by using the canonical formalism. The Faddeev-Po- pov ghost is shown to be a consequence of two requirements : i) the vector states describing physical and longitudinal vector mesons should be orthogonal and ii) there should be no transi t ion between physical and unphysical modes. The Slavnov-Taylor identi t ies are derived. The metric of the vector space is studied.

1. - Introduct ion .

S o m e p r o b l e m s of n o n - A b e l i a n g a u g e theo r i e s can p r o b a b l y b e b e t t e r h a n d l e d

if one u n d e r s t a n d s m o r e d e e p l y t h e u n u s u a l f e a t u r e s of t h e s e f ield t h e o r i e s

f r o m a p h y s i c a l p o i n t of v iew. I n ou r op in ion , for th i s p u r p o s e t h e L a g r a n g i a n

f o r m a l i s m can be a g o o d c o m p l e m e n t to t h e a p p r o a c h e s c o m m o n l y used to fo r -

m u l a t e t h e s e theo r i e s (1), e.g. t h e p a t h i n t eg ra l .

(*) To speed up publ icat ion, the authors of th is paper have agreed to not receive the proofs for correction. (**) Permanent address: I s t i tu to di Fisiea dell 'UniversitY, Pisa. (1) L . D . FADDE]~V and V. N. PoPov: Phys..Lett. , 25B, 20 (1967); G. '~ HooFs and M. V]~LTMA~: Diagrammar, CERN report 73-9 (1973); B. W. L~E and J. ZINN-JuSTIN: Phys. l~ev. D, 5, 3121, 3137, 3155 (1972); 7, 1049 (1973); E. S. FRADKIN and J. V. TYuTIN: Phys. t~ev. D, 2, 2841 (1970).

155

156 G. cuRci and R. FERRARI

The problems t h a t we have in mind are, for instance, the proper t ies of the

spec t rum of the f o u r - m o m e n t u m (already the Abelian gauge theories, wi th a Higgs phenomenon, have an unusual spec t rum (2)), the s t ruc ture of the vector

space where the fields are defined (for instance, its metr ic , the physical sub- space), the s y m m e t r y propert ies of the full theory, the construct ion of observable opera tors (3).

For these problems it is cer ta inly helpful to have a formula t ion where locality and covar iance are formal ly preserved and the ghost field is t r ea t ed a t the s~me level as the other fields (4).

We shall consider only the SU~ gauge group, b u t there are no difficulties in generalizing to more compl ica ted groups. Our formula t ion and results are

valid in cases of bo th exact and spontaneously b roken s y m m e t r y , i.e. the gauge t ransformat ions of first k ind can be spontaneous ly b roken and thus there can be a Higgs phenomenon and the vec tor mesons can be massive.

We shall s ta r t f rom a gauge- invar ian t Lagrang ian and then int roduce a symmet ry -b r eak i ng t e r m propor t iona l to the square of the divergence of the vector field, in order to be able to quant ize the theory canonically and in a Lorentz -covar ian t way. The presence of this new t e r m brings difficulties in sat isfying a t the same t ime bo th the condit ion of or thogonal i ty be tween physical and unphysical vector mesons and the requ i rement t h a t the physical s ta tes do not in terac t wi th the unphys ica l ones. The in t roduct ion of the Faddeev-

P o p o v ghost is shown to solve these difficulties and the S l a v n o v - T a y l o r identit ies are easily proved.

The interact ion t e r m between the ghost and the vec tor -meson field poses some problems for the Hermi t i c i ty of the Lagrangian . I t is shown t h a t there exists a met r ic which avoids this difficulty.

I n these considerations it is useful to follow the paral lel ism of the non- Abelian gauge theories with the Gupta-Bleuler formula t ion of QED.

2. - T h e e q u a t i o n s o f m o t i o n .

We s ta r t f rom the following Lagrang ian densi ty:

1 G2 a

(2) R. FnRRARI: Nuovo Cimento, 19A, 204 (1974). (3) H. KLUBERG-STERN and J. B. ZuB]~]~: Ward identities and some clues to the renor- malization o] gauge-invariant operators, CERN-SACLAY preprint DPh-T/74-56. (4) An approach to this problem with Lagrangian formalism has been used previously by E. RuboLe~ and H. P. Di~RR: Nuovo Cimento, 10A, 597 (1972). Our results are, however, somewhat different from theirs. For instance, we show that the interaction between ghost and vector mesons cannot be given by their eq. (1.18) if the theory is local.

157 A C A N O N I C A L A N D L O I ~ E N T Z - C O Y A I ~ I A N T Q U A N T I Z A T I O N O F Y A N G - M I L L S T H ~ O I ~ I E 8

where

(2) G ~ ~-- ~ A ~ -- ~ A ~ + gA~ × A~,

ju is the (~ minimal coupling ~) current and ~-~g is the Lagrangian of the ma t t e r field (in some cases, e.g. scalar ma t t e r field, bo th ju and ~e~ can contain A~,,

bu t not its derivatives). I f g = 0, Lf is invariant under the following infinitesimal t ransformations:

A~,(x) --> A~(x) + gA(x) ×A~,(x) -- ~ A ( x ) ,

(3) Ze~(x) -> ~e~(x) + j . ~A(x),

j.(x) ~j~,(x) + gA(x) ×i~(x) ,

where A(x) are e-number functions. Under (3) G~ transforms like an isotopic vector :

G,~(x) ->G,~(x) + gh(x) ×G,~(x).

The covariant der ivat ive is defined by

i j i k i k ge A~, (a) D~ -

and it is easy to see t ha t

(5) _ ~ _ ~ -- D~ D z = gG~ .

As is well known, the canonical quant izat ion of (1) with a = 0 is not possible if one wants to t rea t all the Lorentz components of A~ in a covariant way. The t e rm proport ional to a is in t roduced to overcome this difficulty, bu t un- fo r tuna te ly i t breaks the s y m m e t r y (3) unless A is constant .

The conjugate momenta of Au are

(6) ~(A~) = ~ ,~o_ ~g.O ~A'~,

and the equations of mot ion are

(7) (D ~ G~)~ + ~ ~ A ~ = - - j ' , .

By using eq. (5) we have f rom (7)

(8) ( D ~ . ~ A ~ ) ~ - ] - (D~j~) ' = 6 .

158 G. CVRCI and ~. FE~RAm

We now m a k e use of the isotopic-spin invar iance of ~f (A ~ const). F r o m

~ o e t h e r ' s theorem we get the conserved isotopic-spin current

(9) J~, -~ gA~ × G~,- t - ag ~'~ A , × A~, ÷ j ~ .

B y using the conservat ion of J~ and the equat ion which comes f rom the diver-

gence of bo th sides of (7), we have

(10) [] ~. A~ ÷ gA. × ~, ~o A~ = 0,

or, in short nota t ion,

( D , ~ A a ) i ~ O .

We call (10) the equat ion for the supp lemen ta ry condition. I t is wor th not ing its complexi ty (nonlinearity) with respec t to the similar one in QED ([] ~oA~ ~ 0).

Moreover, eq. (10) does not depend on the m a t t e r fields as long as the inter- act ion with the vec tor field is given b y the min imal coupling. On the classical level it would be interest ing to s tudy this equat ion and, in par t icular , to see whether " i A , ~-0 a t any t ime if i t is zero with its first der ivat ives over some space- like surface. The pe r tu rba t ive solution of the quant ized version of this equat ion indicates t h a t it is indeed so.

We conclude this Section with some formal developments . Using the equa- t ion of mot ion (7) we can write the isotopic-spin current in the following way:

T o w G~ is an t i symmet r i c in the Lorentz indices and, therefore, for any local field B we have

f d 3 x [ ~ G~o, ~ O, B]

thus this t e r m of the current does not contr ibute to the isotopic charge and the isotopic-spin current can be chosen to be

(11) J~ (x ) = - - ~¢(D~ ~ A~) ' (x ) ,

and eq. (10) is the s t a t emen t t h a t this current is conserved.

3. - The supplementary condit ion.

Up to now we have been wri t ing the equations in a f o r m a l way and we have used the fields as if t hey were classical. These equat ions will be defined

A CANONICAL A ~ D L O R E N T Z - C O V A R I A N T QUA~TIZATIO:N OF YANG-MILLS TId]!~ORIES 159

in the quan tum version of the theory through th e per turba t ive expansion. In part icular , we shall assume tha t our fields have asymptot ic limit and tha t asymptot ic states are complete. Moreover, the T-product is required to be a covariunt quant i ty .

:Now we consider the case where the vector field describes asymptot ica l ly massless vector mesons plus some unphysic~l states. A vector meson is then described by a funct ion ~vs(x) which should be transversal:

(12) [ ] i ~.(x) = ~" ~ ( x ) = o .

tt t Any other funct ion ~v~ which differs f rom ~v, for a gradient of ~ scalar funct ion should describe the same state (as in QED)

(13) 'i i Zi Xi . F~ = ~f, -F ~, with [] ~ 0

The state of a vector field is then described by a class of equivalence defined by the relat ion (13). This fact can be expressed by the equation

(14) <%= I~.A~ (x)l~i= > = o , ou~ o u t OUt

where qs and ~p describe any physical states. Equa t ion (14) is the supplemer~tary condition defining the physical subspace.

The same condition should be valid also in the presence of a t t iggs phe- nomenon. In this case (2) the asymptot ic field can be split uniquely into two par ts : one field is t ransversal and describes a m a s s i v e vector meson, the diver- gence of the other field describes unphysical states which should be orthogonal to all physical vectors. Then the same condition (14) is valid for the physical subspace.

To have a theory which has a physical interpretat ion, eq. (14) is certainly not enough. I t should be asked also tha t the longitudinal massless vector- meson states are orthog0nal to a n y physical states. In par t icular it must be

(15) ~(+~ ~.Aoo, ( x ) l ~ > = 0. la o~t

The requirement tha t for any physical vectors ~ and

(16) <~I~A~,(x) I~P> = O

is cer tainly a su]] ie ient condition for the val idi ty of bo th eqs. (14) and (15). Equa t ion (16) is not compatible with eq. (10) derived in the previous Section.

In the nex t Sections we shall s tudy a per turba t ive solution of eq. (10) and see how the theory can be modified in order to satisfy eq. (16).

160 G. CURCI and ~. FF~RRARI

I n QED ecl. (16) is necessary also to preserve the Maxwell equa t ion of mot ion (5), bu t in Abel ian gauge theories there are no problems, since the con- servat ion of e lect romagnet ic current implies [] ~o A~ = 0 and, therefore, eqs. (14)

and (15) are consequences of each other. I n non-Abelian gauge theories condition (16) is not easily connected with

the va l id i ty of classical equat ions of mot ion (7). I n fact we shall see t h a t the in t roduct ion of the F a d d e e v - P o p o v ghost changes the classical equat ions of motion. On the other hand, the equat ions are not l inear in the vec tor fields and so, even if we t ake the expec ta t ion value between physical states, there can also be contr ibut ions f rom unphysica l vectors in the in te rmedia te states, which

could cancel the ghost te rm.

4. -Perturbative solution of the equation for the supplementary condition.

We solve eq. (10) pe r tu rba t ive ly in g using Schwinger 's formalism. We

add to the Lagrangian (1) an externa l source t e r m

(17) - - h~(x) . A~(x) ,

and consider the funct ional

The Green 's functions (disconnected) can be genera ted f rom this funct ional in the usual way b y means of funct ional der ivat ives with respect to t h,(x) .

Equa t i on (10) changes because of (17). The isotopic-spin cur rent is no

longer conserved, in fac t

(19) ~ j ~ = ~Lf ~, ~ ~ ~A i - gA ht, e .

Proceeding as in Sect. 2, f rom the equat ion of mot ion and f rom (19) we have,

instead of (10),

(20) [] ~ A ~ ÷ g ( A . × ~ ~A~) = 1= (~.h~ + g A . × h~,) .

For la ter use we wri te the second t e r m on the lef t -hand side in a different

fo rm:

(21) [] ~ , A , ~ - g[(1 - - ~ )A, × O,O~Aa -t- ~O,(A~ × O~A,)] = 1 (~,h~,@ gA~' × h , ) . 6~

(~) In fact, this condition has remarkable consequences in a local theory of QED. See R. F ~ R ~ I , L. E. PICASSO and F. ST~OCCJaI: Comm. Math. Phys., 35, 25 (1974); F. STROCeHI and A. S. WmHTMAN: Prooj o] the charge superselection rule in local rela- tivistic quantum ]ield theory, Princeton University prcprint (1974).

A C A N O N I C A L AN]:) L O R E N T Z - C O V A R I A N T Q U A N T I Z A T I O N OF YA/~IG-MILLS T H E O R I E S 161

Now we t ake the expectat ion value of (21) between in- and ou t -vacuum and we perform functional derivatives with respect to the external sources• For instance the first t e rm of the l.h.s, of eq. (21) gives

(21a) 8nEJ~<O°ut lA~(x) lOin> l D ~ u ~ ( O t T ( A ~ ( x ) A ~ , ( y l ) i~ . . . . . . . . . Azn(y,)10 } . (4) ~h~,,(yl) .•. ~h~,,~(y~) ]h~

Finally, we t ransform (21) into an integral equat ion using the Green's function D~ ( D D~(x) = - ~,(x)):

(22) ( O I T ( O A, , (x )A~, , (ya) .. A,Jy~)) t0 } ' y D , ( x - - y ) .

• <olI_gT ( , , . . , ° A , , ( y ~ ) . A~, : (y~)s '~k [ ( 1 - - 2 ) A ~ ( y ) ~ " A ~ ( y ) ~- 2~"(A~(y)~"A~(y))]) ~-

+ i-~ ~ o ,, ey--; ~(y-- y~)T(A2(yl)... A2(y~)... A,Jy~)) +

- - ~Ji.~ .. . A ~ " J y , ) ) ]0) , J

where ~ - -means omitted. In order to simplify the expression in eq. (22), we have kep t the derivatives of the vector field inside the T-product . In fact, like in eq. (21a), they should appear outside. Thus if we insist with the convention used in (22), we must discharge the noncovar iant terms t h a t one obtains in taking the derivatives outside the T-product . I t can e~sily be seen tha t the first t e rm on the r ight-hand side of eq. (22) is not covariant . In fact, taking the derivatives out of the T-product , one obtains noncovar iant terms. Then the T-product should be defined as if the two derivatives were outside the T-product .

I t is useful to write (22) in momen tum space and to represent the equation graphically. M is the nont runca ted and disconnected ampli tude

{ f d , p . ~ . . . . . o (23) " ~ M iil'''i" t" =- i - - g (--~)4 e [ ( i - - z ) p + 2qe]"

q M i k i l . . . i n ~ 1 n

M ~ ' " ~ ' " i " ~ • q r ~ . . . , ~'~ ~=i • , , . .~ , . .~ , , tq~ , . . , q , , ) - - _ _ ~ , e ,,~,,...'~,...~,Aq, ql , . . . , ..

(2~) = =

D t]

_ ~ I .-""-'" "~'* 1 r 1 ~ f ~ t~=~ D

11 - 11 N u o v o C imen to A .

162 G. o u R e i &n4 R. F E ~ R A R I

If all the momenta are incoming, the graphs mean

(25)

q i ' M ~ , . . . . . ~ # - ql , , q oc

i "Jq

q~ + ie ' " " -~ i/~ - - g ~J"g,, , Ivp OC~ q

7q- - - < J i ~ __ geij~ [(1 - - 2)Pl ,P~-- 2q~,P,]"

P

Moreover, the F e y n m a n rules eoming f rom the Lagrangian (1) give the vertices and the propagators for the vector mesons and the other particles.

Any of the eqs. (22), (23) and (24) can be i terated. To a fixed order in the power of the coupling constant g we have

+ ~: . . . . . { . . . . . ) _ _ - . ~ + : - -~- . . . . p _ ~ . ,.. • r = 1 ,

r=l ~ ~ n ~ n The loop of the massless scalar appears only once in each graph of the type

shown in the first two terms on the r ight-hand side. The scalar interacts with the vector meson through the ver tex (momenta are incoming)

(27)

v - ~'~- - / - ~ / ' / J P

Iq ,~, kp x \

- g ~ , , ( - ( 1 - ~)p + ~q), .

The following two vertices appear only once in each graph: p.

/ t j

/ /

(28) -.-b.._--~ / -g s'Jk(-- ( 1 - - ~t)p --~ - ~ q ) ( q - - ~ - p ) ,

kp \ . \

(29) :. - , . - - - 2 tq . ,,%

A CANONICAL ~ N D L O ~ N T Z - C O V A I ~ I A N T QUA~ITIZ&TION OF YANG-MILLS T H e O r I E S 163

The short double line in the th i rd graph of (25) and the arrows in (27), (28) and (29) keep t rack of the fac t t h a t these graphs are not symmet r i ca l in the

scalar legs (these vert ices do not come f rom Lagrang ian terms).

F r o m eq. (26) we see t ha t the scalar loops are responsible for the violat ion of eq. (16). I n fact , if we app ly the reduct ion formulae to eq. (26), we see t h a t the th i rd t e rm is zero since the polar izat ion of a physical vec tor meson is t rans- versal (eq. (14)) and the four th is also zero because there is no pole in the ex- te rna l r - th m o m e n t u m , t towever , in general the first two classes of graphs give nonzero contr ibut ion. I f we also reduce the externa l leg which carries m o m e n t u m q, we see t h a t longi tudinal vector mesons in terac t with physical

s ta tes and so eq. (15) cannot be valid. I t is amusing to see t h a t in the l imit of zero loops (classical limit) eqs. (14)

and (16) are compat ib le .

We shall see in the nex t Section t h a t we can introduce a new scalar field

(the F a d d e e v - P o p o v ghost), coupled to the vec tor field th rough the ver tex (27), which gives loops ident ical to those appear ing in eq. (26). Then if the ghost obeys F e r m i statist ics, the loops car ry an ex t ra fac tor minus one and they can cancel the first two te rms in eq. (26).

5. - Faddeev-Popov ghost and Slavnov-Taylor identities.

We introduce a new scalar field wi th Fe rmi statist ics into the Lagrangian (1). The coupling with the vec tor field is fixed b y the ve r t ex (27) and the coefficient of the kinet ic t e r m b y the va lue of the p r o p a g a t o r

(30) 5e = ~x -- 8~¢9" • 8~ ¢p + gA.[(1 - - 2) 8.q)* xq) - - 2q)* x 8.q) ] .

The in teract ion be tween q) and An is not the min imal coupling fo rm; then eq. (10) is modified.

The equat ions of mot ion are

(31) (D"G,~)i-[ - ~ ' A ~ + g~Jk[(1 - - 2 ) ~ j . q k _ 2 ~ . ~ ] = _ j ~ ,

(32) [] ¢p + g(1 - - 2) ~(A~, xcp) = -- g2A~ × ~ ,

(33) [] ¢.fl* -t- g2~(A~, × ¢?*) ---- - - g(1 - - 2) Ag X ~zcp*.

The conserved isotopic-spin current is

(34) J.=gG~,.xA~+g~8~A~xA.+j~--g(c~*x~,~--8~*×~)--

- - g212(cp* ×A~) × ~ + (1 - - 2)¢~* × (A~ × ~ ) ] ,

since ~o -->¢p + gA×%o.

164 G. CURC~ and a. FERnAR~

With the usual procedure we have

(35) ~(D ~ A ~ + g A . × ~ . ~ . A o ) - - g ( ( 1 - - ~ ) q J * x [ l c ~ - - ~ [ ] ~ * x q ~ +

+ (1 - 24) ~ * x ~q~) --g~ ~,[~(~* xA~) x $ + (1 -- ,~),~* x (A~ x ~ ) ] = 0 .

The quant izat ion of the free scalar field with Fermi statistics presents no

difficulties. The paper by lCUDOLI"H and Di2gR (~) gives the relevant details.

Here we recall only some points. The field can be expanded in plane waves

(36) q~i(x)= (2:~)-Jf d2--qqo(ai(q)exp[--iqx]+ bi*(q)exp[iqx]) ,

a and b annihilate the vacuum and they an t icommute with each other and with themselves; moreover

J {a'(q), aJ*(q')} = 2qo 5 ~j ~3(q_ q,) , (37) / (b'(q), bJ*(q~)) = -- 2qo O'J 5a(q -- q ' ) .

The energy can have positive spectrum only if the metric is negative, due to

the minus sign in the second commuta to r in (37):

(38) <aqilaq'j> = -- <bqilbq'j> = 2% (~ (q - q') ~ .

For later use we shall call ~ the metric given by the canonical commuta t ion relation for the field A~ (for ~ ~ 1 the metric is the same as the Gupta-Bleuter 's one ; for a =/= 1 the si tuation is a little more complicated but its precise form is

not needed here) and by the scalar product in (38). The propagator is the same

as in the normal massless scalar field, but

(39) <0lT(~(x) ~J*(y))10> = -- <0 IT(q~*(y)qS(x))10>.

A remarkable proper ty of this scalar Fermi field is tha t the theory is invariant

under the R-transformat ion defined by

(40) q~(x) -~q~*(x) , q~*(x) ~ - - , . p ( x ) ,

i.e.

a-~>b, b ~--~--a,

a* ~ - ~ b* b* -~ a* , *

Now we can find out what the value of ~ should be. Equa t ion (35) can be solved per turbat ively as we have done in the previous Section. The situation

A CANONIC_~L AND LORESITZ-COVARIANT QUANTIZATION OF YANG-MILLS TI tEORIES 165

is more complicated since here we also have the ghost field and so the graphs

must contain it in all possible ways. The ghost loop will cancel the scalar loop of eq. (26) only if eq. (35) can yield

vertices analogous to (28) and (29) where two scalar legs are replaced by ghost lines. The second te rm in brackets of eq. (35) should give thethree- l ine ver tex ; however, we must be careful in evaluat ing the order in g since the &Alembert ian carries an extra g through the equations of mot ion (32) and (33).

The solution of the problem is simplified by the fact tha t the ve r tex (28) should correspond to a ver tex ghost scalar

. . . . . . g_ #¢k(p. t -+- ~.t~), (41) ;q

where all momenta are incoming. Comparing this with the third monomial in the second te rm in brackets of eq. (35), we see tha t we should have

(42) 2 = o .

Using eq. (32) we get f rom (35)

(43) g2

[] ~ A ~ + g A u x ~ ~ A ~ -- gg- Outp* x ~ ,q ) - - - - ~uq)* x (A~ x (p) =- O .

The last t e rm gives a ver tex

ig ~ ~ikn zn~jp t6 (44) - , ~ - - -

kp-~,

which is in agreement with (29). Finally we see tha t the ghost loop cancels the scalar loop and eq. (26) be-

comes (Slavnov-Taylor identities)

(45) . . . . . . . . ,

p=l ~ r'=1

where the amplitudes are buil t using the F ey n m an rules coming from the La-

]66 G. CuRcI and R. FERRARr

grangian (30) with 2 = 0. The scalar line interacts only wi th the vec tor line through the ver tex (27). I n this t heo ry eq. (16) is satisfied.

Equa t ion (45) can easily be generalized, using funct ional formalism, to the case where also other part icles are coming out besides the vec tor mesons.

6. - The m e t r i c o f the vector space .

The in teract ing Lagrangian in (30) is not Hermi t i an ; in fact , if we write

(46) g An. £ f ~ , = - - ~ - 8.(q~* x ~ ) ,

i t can easily be seen t h a t the first t e r m is Hermi t i an , while the second is anti- t t e rmi t ian .

We shall show t h a t this causes no t rouble if there are no external ~-lines

in the ampli tude, as for instance in eva lua t ing the S-mat r ix be tween physica l vectors.

The H e r m i t i a n p a r t of (46) is even under R- t ransformat ion , while the anti- ] t e rmi t i an one is odd. Moreover, under R the arrow of a ~ -propaga tor is reversed. Then, since for every g raph with a ~-loop there is the corresponding one wi th reversed arrows, the an t i -Hermi t i an t e r m of (46) can appear only an even number of t imes in every loop. Thus the non-Hermi t i c i ty of ~ea is harmless for Green 's

functions wi th no externa l ~-lines. As in QED, the physica l vectors fo rm a subspace V' c V of the whole vector

space where the fields are defined. A vector ~ is physical if it obeys the sup- p l e m e n t a r y condit ion

(14') ~ A~= (+~ ~ = 0 , out

and it does not contain a n y ghost. On V the t ime evolut ion opera tor is not un i t a ry since the Lagrang ian is

no t t t e rmi t ian . However , this gives no inconsistencies for physical expecta t ion values of the S-matr ix , as we have seen before. Moreover, we shall show t h a t the vec tor space V can be equipped with a scalar p roduc t which renders t t e r - mi t i an the Lagrangian (30).

We define au opera tor 9 which implements the t r ans fo rmat ion R given b y (40). On the a sympto t i c s ta tes 9 is un i ty on the vec tor mesons and on the ghost s tates we h a v e

(47)

~1~> = - Ib>,

~Ib> : [a>,

and also

elo> = 1o>.

A CANONICAL AND LORENTZ-COVARIANT QUANTIZATION OF YANG-MILLS THEORIES 1 6 7

Therefore

~ 2 = - - 1

on the one-ghost s tate and ~ is Hermi t ian under ~-conjugation:

~*---- ~.

In V we can define a ~ scalar p roduc t

which gives a new nondegenerate metric. The t t e rmi t i an conjugation in the ~-metrie is then

(49) B ~ = c 1 B * ~ ,

where B* is the ~]-Hermitian conjugate of B. The Lagrangian (30) is t t e rmi t i an in the ~-metrie. In fact we have

(9~*)t = ~-~9~ = ~o*,

(A~) + = A~;

therefore bo th terms in (46) are even under ~-Hermit iaa conjugation. Wi th the ~-metrie the theory remains local and Lorentz covariallt, at least

a t the level of per turba t ive expansion.

7. - Conclusion.

We have given a Lorentz-eovar iant and local formulat ion of the Yang- Mills theory, t reat ing physical and ghost fields at the same level. Moreover, we have to just i fy the presence of the ghost f rom some physical requirements.

We hope tha t some problems can be solved more easily with our approach. For instance, we must prove tha t the unphysical states do not enter into the physical expectat ion values of the S-matr ix (proof of unitar i ty) .

I t would be interest ing to s tudy in our approach the symmet ry propert ies of the theory and how the s y m m e t r y t ransformations are implemented (6)

(s) In this respect there has been some appealing suggestions by C. BEeCHI, A. Rol~v.¢ and R. STORA: Abelian Higgs-Kibble model. Unitarity o] the S operator, Marseille pre- print 74/P.625 and l~e~ormalization o] the Abelian Higgs-Kibble model, Marsdlle pre- print 74/P.634.

168 G. CURCI and R. FERRARI

and , moreover , w h a t the consequences are of the fac t t h a t the phys i ca l vec tors

are classes of equ iva lence wi th nonde f in i t e i so topic-sp in charge (see eq. (13)).

This second p o i n t b r ings us to the p r o b l e m of def in ing obse rvab le q u a n t i t i e s

in n o n - A b e l i a n gauge theories . To our knowledge this p r o b l e m has n o t ye t

b e e n considered.

* * *

One of the au tho r s (RF) wou ld l ike to t h a n k Prof . A. SALAd, the I n t e r -

n a t i o n a l A t o m i c E n e r g y A g e n c y a n d U N E S C O for h o s p i t a l i t y a t t he I n t e r -

n a t i o n a l Cent re for Theore t i ca l Phys ics , Trieste .

• R I A S S U N T O

Usando il formalismo canonico si quantizzano le teorie di (( gauge ,~ non abeliane in modo covariante. Si dimostra che la presenza del (( ghost ~> di Faddeev e Popov ~ una eonseguenza di due richieste fisiche: i) gli stati che deserivono mesoni vettori fisici e longitudinali devono essere ortogonali e ii) non ci deve essere transizione tra vettori fisici e non fisici. Si derivano le identit~ di Slavnov e di Taylor e si studia la metrica dello spazio vettoriale.

K a H o m t q e e g o e n 3"[opeHTU-goBapHaHTHOe KBaHTOBaHHe Te0pHfi ~ l u r a - M H ~ e a .

Pe3mMe (*). - - HcrtonJ~3y~t raHonrlqecrH~ qbopMaan3M, rrpoBo~nTC~l flopeHTt~-rOBa- pnaaTHOe KBaHTOBaHHe Hea6eneBs~x KanH6pOBO~HBIX TeOpH~. YIoKa3bmaeTcfl, HTO << ~yx >> dPa~eeBa-l-[ouoBa maa~¢TC~ C~e~CTBHeM ~ByX Tpe60Ban~: l) Be~Topl, I COCTO~IHHk[, ormcbmamm~e ~bn3n~ecKHe n IlpO~OJIbHble BerTopnbIe Me3OHbI, ~OII;~HBI ~bITb OpTOrO- Ha~bHbI, 2) ne ~ o ~ n o cymeCTBOBaT/, ilepexo~oB Me~Iy ~bH3HxleCI(HMH H neqbn3rr~ecrnMn MORaMI~. hbmoRflTCa ToxReCTBa C~aaHoBa-Te~opa. HccaeRyeTc~ MeTprrKa BeKTOp- HOrO ~pocTpaHCTBa.

(*) HepeaeJeno pec)amtueh.

by SocietA Italiana di Flsica

ProprietA lotteraria riservata

Direttore responsabile: CARLO CASTAGNOIJI

Stampato in Bologna dalla Tipogr~fla Compositori coi t ipi 4ella Tipografla Monograi

Questo fascicolo ~ stato licenziato dai torvhi il 27-X-1975