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Physical unitarity in the BRST approach
Volume 217, number 1,2
PHYSICS LETTERSB
19 January 1989
PHYSICAL UNITARITY IN THE BRST APPROACH A.A. SLAVNOV
Steclov Mathematical Institute, Moscow, USSR Received 4 August 1988
The conditions of unitarity in the positive subspaceof an indefinite metric theory are given. BRSTquantization is considered as the most important example.
The covariant formulation of the most physically interesting models requires the introduction of an indefinite metric space. One of the most popular examples is given by the BRST quantization of gauge theories [ 1-5 ]. In this approach the S-matrix is defined in a large space including ghost states, and the physical subspace is separated by the condition QIO)=o,
(1)
where Q is the BRS charge. The operator Q commutes with the hamiltonian, therefore the S-matrix is obviously unitary in this subspace. However, it is by no means evident that the states (1) are really "physical", i.e. have a non-negative norm. For the Yang-Mills theory this fact was proven in refs. [ 1,2 ]. There exists also a proof of the equivalence of the BRST quantization and the Dirac quantization for the gauge theories with independent constraints and closed algebra [6,5]. However, in the general case when the constraints are dependent and the algebra may be open the proof of the positivity of the physical states norm was absent. In this paper we formulate the conditions providing the unitarity of the S-matrix in the positive norm subspace. No assumptions about the gauge group and the existence of a unitary gauge is needed. In fact our proof works equally well when there is no gauge invariance at all. We consider a theory described by the Lorentz invariant non-degenerate lagrangian ~(Ai, c~, dg). Here c~, gg are ghost and antighost fields, and the index ~ denotes ghost number. All the fields are her-
miti~an. We assume also that the free lagrangian is quadratic. The corresponding creation and annihilation operators satisfy the standard (anti)commutation relations [c~(k), ~ ( k ' ) ] + = g ~ 6 ( k - k ' ) , g~ =diag{+ 1.... , + 1, - 1,..., - I}.
(2)
Analogous relations are valid for the operators as(k), aj(k' ). The fields with ghost numbers differing by 1 have an opposite Grassmann parity. There exists an operator Q(a~, aT, c~, cf*, :'~ c~*) with the following properties: ( 1 ) The operator Q allows a representation
Q=Qo+O(g),
(3)
where Qo is quadratic in the fields and O(g) denotes higher order terms vanishing when the interaction is switched off. (2) The (anti) commutator of Q with any observable is equal to zero. (3) The operator Q has a ghost number equal to 1. (4) Q is nilpotent,
[Q,Q]+=2Q2=O.
(4)
(5) For any linear combinations ?7=~jb~j(k) ×c~(k), g,~ = ~jb~j(k)~(k) either [Qo, gj~] + ¢0, or [Qo, 27] + ¢0. (6) The operators a* (k) (anti) commuting with Qo are physical, that is the states generated by these operators have a positive norm. 91
Volume 217, number 1,2
PHYSICS LETTERS B
19 January 1989
If the operator Q satisfies the conditions ( 1 ) - ( 6 ), the following theorem is valid: The S-matrix constructed with the help of the lagrangian is unitary in the subspace ( 1 ) and the corresponding asymptotic states have a non-negative norm. First of all let us note that for asymptotic states condition ( 1 ) is simplified:
It follows from the nilpotency condition (4) that
Ql¢)=0oQo[0as) =0.
a+ _ ai +bdi1. ad a , - b a i1. ad X/~ , a;-x/~ g,;,
(5)
Here we have used the assumption about linearization of the asymptotical dynamics which is valid in our case at least in the framework of perturbation theory. One can easily verify that the operator Qo commutes with the S-matrix [ 7 ] and therefore the S-matrix is unitary in the subspace of asymptotic states satisfying eq. (5). Now we shall prove that this subspace has a non-negative norm. The most general form of the operator Qo which is quadratic in the fields and has a ghost number equal to 1 is
Qo=
f dk
~ a*(k)fl~((k)c)(k) \i=lJ=l
N2 ~12
t
o~=2 3= I / = 1
(6)
?~={cT-,d~-},
i=l,...,N,,
j = l .... , n , ,
d = n ~ + l .... , N ~ ,
(7)
where th~
o~ ot-6°J- = Z bdjcj .
(8)
j=l
In terms of these fields the operator (20 looks as follows:
ncc + 2 92
= (a 7 +a~bdj)C))
~. (e7 ~ - ' + e 3 ~ - ' b g j ) c 7 - + h . c . .
~=2j=l
[c) - , ~ '~ +Cd -*' bdk]+ ~- =o ,
(ll)
etc. Here for definiteness we assumed that the fields a, have Bose statistics. Eq. (10) leads to the following commutation relations for the fields:
(9)
(12)
[a +, (a+) *]=[a#, (a7)*]=0, [a+ (k), ( a j ) * ( k ' ) ] = ~ j ~ ( k - k ' ) .
(13)
The first term in the equation for Qo may be written in the form
QO=
dk
[(a+)*c)-(c)-)*a +]
(14)
If n~ is equal to the number of the original operators then due to the independence of c)-, the operator Q°o acts non-trivially on all the ghost fields ?] (k). The matrix fl'? has an inverse (15)
and one can introduce creation operators (d~+)*= ([3F')k:(d])*
Let us introduce linear dependent fields ~ 7 = ~ f l , c / . If the maximal number of their independent components is n,, then the fields 8~ may be represented as follows:
I dk
(10)
ff{g~k(flf~)k:= 6",
+ E E Z eJ'~-~(k)fl~(k)c'~(k) +h.c. •
Qo=
[a/+b~Tad, a~ +a*b~,k] = 0 ,
(16)
such that [c~-(k), (0+)*(k')]+ =6,j~(k-k').
(17)
The operator QO acts in the space spanned by the vectors generated by the operators (a +)*, (a j ) * , ( c ] - ) *, ( (~ + ) * and satisfies the conditions ( 1 ) - ( 6 ). In this case only one generation of ghosts is needed. This situation is realized in the Yang-Mills theory and in other models with irreducible constraints and closed algebra. If the number of independent operators is less than the original number of ghosts (n~ < M~ ) the operator Qo° does not act on some of the fields c) and to construct the operator satisfying the conditions ( 1 ) - ( 6 ) one needs higher ghost generations. The operator Q°o is constructed in the same way as before, the creation operators ((~-+)* being defined by eq. (16) where (fli-~)k: is a right inverse matrix. The next term in Qo looks as follows:
Volume 217, number 1,2
(c, +Cd bdi)C~ \~=
PHYSICS LETTERS B
+h.c.
dk.
(18)
=
(]
Qo= f d k
1
The operator Q~ annihilates all the vectors created by ( a + ) *, ( a 7 ) * , (c~-)*, (?)+)* due to condition (It). The operators ~)+
19 January 1989
- , _= yi/Cj-~ + b,2 d .Cd
(19)
are independent and one may introduce creation operators
Y. [ d * ( k ) c , ~ ( k ) + c * ( k ) d , ( k ) ] ,
O~=1
(26)
where N is the total number of different types of partides. In particular N may be equal to oo. An arbitrary state vector may be represented as a sum of the vectors
IO ) = [ O ) p ~ ® l ~ )
,
(27)
where I¢)ph is a positive norm state, and the vector Ig~) looks as follows:
(~,!)*=d*(Y-')~,, I ~)=
[() + (k), ( ( ) - ) * ( k ' ) ]+ = 8 0 ~ ( k - k ' ) .
E
--10~l'l~'
]""
OL\Pn,,]
O!'Lll ) ' ' "
.(qm.)
(20) * " )...c~(sk.) * " ( *. ( t ", )... ( *4 ( /t ". ) xc~(s,
Defining in the same way (c}+). = (2. (fl~-,)/~,
tll...ll,Pv',llll...t~lN ...... S l ...S ~,,, t ', ...t /.,, nx, q I ...q N × Gk,...k,,../,..Z~ (P l1...p JV x)
(21)
Xdp', ...dt~. ,
(28)
we get the following representation for the Q~:
Q~= fdk(~=l(~]+)*c~-+(c2-)*~)+ ) .
(22)
This operator acts in the space spanned by the operators (~)+)*, ( ~ - ) * , ( c ~ - ) * , ((~+)*. Again if n2 is equal to the number of the original operators c~ then the series in (9) terminates, otherwise we must consider the next term, etc. Therefore the general structure of the operator Qo is given by the following expression:
where the functions G are symmetric with respect to the arguments p,", q~ for fixed c~, and antisymmetric with respect to s~, t~. Substituting the vector (2 7 ) into eq. ( 1 ) we get the following relation between the functions G: [ G")--"~'"n'~,m~-"mB--'m,xt~ I /3
k) ...klJ...k~,Ib..I/j...l,v
N q lt ...q N,,,x, S ~I ...S kx,
dk
t
~ (a~-)*c]-+ I
~=1
(g]+)*c]-
~ G n l . . . n B + I...tlA,ml...m/3+ l...m~¢ [ ~ fl-- 1 kl...k B -- 1...k:VJl .../fl-- 1...IN ~1j I " ' P n/~- t tfl~fll/ / 1 ""//nN,~N
I ~l~fl l a l t l l . . . t l~mNN ,
t l ....+#,_,-2 f l - l ~fl ....+/xx ) = 0 G ll [...O...pl.N.ill [...lVll%
+ ,=~ (~,~+)*c3- + . . . + h . c . ) ,
(23)
where each term operates in a separate subspace. Now we can study the problem of positivity of the physical states norm. It is convenient to change notations. In the following we shall denote all Bose operators as a~, d,, [ a . ( k ) , d*~(k') ] = 6 . a 8 ( k - k '
),
(24)
and all Fermi operators as c., c~., [c,~(k), e~(k') ] + = ~ a ~ ( k - k ' ) .
t I ... t )x) A,~
+ (m/~+ 1 )( -- 1 ):;~_, k,+~_~/.
q,1 . . . ,n, i, l, -
Qo=
(25)
In these notations the operator Qo looks as follows:
~/~ I¢
~II I " " P n x )
~,.k,,/~_lx
s lI...S~k[]_ It s2/3...S~, ,
(m,,l/~0)
[...Itl~ =Gk,]l [...]~1%'.t?l o ~x.I, Iu=O.
, (29)
In this equation it is understood that the functions G are symmetrized according to the corresponding Young tableaux which can be easily deduced from the representation (28). If we now calculate the contribution to the norm of the terms with occupation numbers (n~...na...nN, ml ...m/~...mN, kj ...kl~...kN, Ii ...lfl...lN) and (nl ...na + 1 ...nx, m~ ...rn/~ + l...mx, k~ ...k/~- 1...kN, Ii ...[~-- 1.../N) and take into account relation (29) we find that these terms have the same absolute value but differ by a sign. Therefore they cancel in the sum and only the terms with zero occupation numbers give a non-zero contribution. 93
Volume 217, number 1,2
PHYSICS LETTERS B
An arbitrary state vector satisfying eq. ( 1 ) may be represented in the form
10) : 10)phG 10o),
(30)
where I0 ) ph has a positive n o r m a n d I0o) has a zero norm. The vectors 10o) are orthogonal to all other vectors. Hence we may identify all the vectors differing by ]~o). The positive n o r m physical space is the subspace ( 1 ) factorized with respect to the null vector space ~o. So we proved that if the theory satisfies the conditions ( 1 ) - (6) the S-matrix is unitary in the positive n o r m subspace and physical unitarity holds. No assumptions about the gauge group were needed. The only i m p o r t a n t limitation is the hypothesis of asymptotic dynamics linearization. It is valid in standard
94
19 January 1989
field theory at least in the framework of perturbation theory. However, in some models like first quantized strings it may not be true. The conditions ( 1 ) - ( 6 ) are met in a natural way in the BRST approach to gauge theories. Some examples of this procedure a n d a more detailed discussion will be given elsewhere.
References [ 1] T. Kugo and I. Ogima, Suppl. Prog. Theor. Phys. 66 ( 1979 ) 1. [2] G. Curgi and R. Ferrary, Nuovo Cimento 35A (1976) 273. [ 3 ] I.A. Batalinand G. Vilkovisky,Phys. Rev. D 28 ( 1983) 2567. [4 ] I.A. Batalin and E.S. Fradkin, Phys. Lett. B 122 ( 1983 ) 157. [5] M. I-]enneaux,Phys. Rep. 126 (1985) 1. [6] R. Marnelius, Phys. Lett. B 99 ( 1981 ) 467. [ 7 ] L.D. Faddeevand A.A.Slavnov,Gaugetheories; introduction to quantum theory (Benjamin, New York, 1980).
PHYSICS LETTERSB
19 January 1989
PHYSICAL UNITARITY IN THE BRST APPROACH A.A. SLAVNOV
Steclov Mathematical Institute, Moscow, USSR Received 4 August 1988
The conditions of unitarity in the positive subspaceof an indefinite metric theory are given. BRSTquantization is considered as the most important example.
The covariant formulation of the most physically interesting models requires the introduction of an indefinite metric space. One of the most popular examples is given by the BRST quantization of gauge theories [ 1-5 ]. In this approach the S-matrix is defined in a large space including ghost states, and the physical subspace is separated by the condition QIO)=o,
(1)
where Q is the BRS charge. The operator Q commutes with the hamiltonian, therefore the S-matrix is obviously unitary in this subspace. However, it is by no means evident that the states (1) are really "physical", i.e. have a non-negative norm. For the Yang-Mills theory this fact was proven in refs. [ 1,2 ]. There exists also a proof of the equivalence of the BRST quantization and the Dirac quantization for the gauge theories with independent constraints and closed algebra [6,5]. However, in the general case when the constraints are dependent and the algebra may be open the proof of the positivity of the physical states norm was absent. In this paper we formulate the conditions providing the unitarity of the S-matrix in the positive norm subspace. No assumptions about the gauge group and the existence of a unitary gauge is needed. In fact our proof works equally well when there is no gauge invariance at all. We consider a theory described by the Lorentz invariant non-degenerate lagrangian ~(Ai, c~, dg). Here c~, gg are ghost and antighost fields, and the index ~ denotes ghost number. All the fields are her-
miti~an. We assume also that the free lagrangian is quadratic. The corresponding creation and annihilation operators satisfy the standard (anti)commutation relations [c~(k), ~ ( k ' ) ] + = g ~ 6 ( k - k ' ) , g~ =diag{+ 1.... , + 1, - 1,..., - I}.
(2)
Analogous relations are valid for the operators as(k), aj(k' ). The fields with ghost numbers differing by 1 have an opposite Grassmann parity. There exists an operator Q(a~, aT, c~, cf*, :'~ c~*) with the following properties: ( 1 ) The operator Q allows a representation
Q=Qo+O(g),
(3)
where Qo is quadratic in the fields and O(g) denotes higher order terms vanishing when the interaction is switched off. (2) The (anti) commutator of Q with any observable is equal to zero. (3) The operator Q has a ghost number equal to 1. (4) Q is nilpotent,
[Q,Q]+=2Q2=O.
(4)
(5) For any linear combinations ?7=~jb~j(k) ×c~(k), g,~ = ~jb~j(k)~(k) either [Qo, gj~] + ¢0, or [Qo, 27] + ¢0. (6) The operators a* (k) (anti) commuting with Qo are physical, that is the states generated by these operators have a positive norm. 91
Volume 217, number 1,2
PHYSICS LETTERS B
19 January 1989
If the operator Q satisfies the conditions ( 1 ) - ( 6 ), the following theorem is valid: The S-matrix constructed with the help of the lagrangian is unitary in the subspace ( 1 ) and the corresponding asymptotic states have a non-negative norm. First of all let us note that for asymptotic states condition ( 1 ) is simplified:
It follows from the nilpotency condition (4) that
Ql¢)=0oQo[0as) =0.
a+ _ ai +bdi1. ad a , - b a i1. ad X/~ , a;-x/~ g,;,
(5)
Here we have used the assumption about linearization of the asymptotical dynamics which is valid in our case at least in the framework of perturbation theory. One can easily verify that the operator Qo commutes with the S-matrix [ 7 ] and therefore the S-matrix is unitary in the subspace of asymptotic states satisfying eq. (5). Now we shall prove that this subspace has a non-negative norm. The most general form of the operator Qo which is quadratic in the fields and has a ghost number equal to 1 is
Qo=
f dk
~ a*(k)fl~((k)c)(k) \i=lJ=l
N2 ~12
t
o~=2 3= I / = 1
(6)
?~={cT-,d~-},
i=l,...,N,,
j = l .... , n , ,
d = n ~ + l .... , N ~ ,
(7)
where th~
o~ ot-6°J- = Z bdjcj .
(8)
j=l
In terms of these fields the operator (20 looks as follows:
ncc + 2 92
= (a 7 +a~bdj)C))
~. (e7 ~ - ' + e 3 ~ - ' b g j ) c 7 - + h . c . .
~=2j=l
[c) - , ~ '~ +Cd -*' bdk]+ ~- =o ,
(ll)
etc. Here for definiteness we assumed that the fields a, have Bose statistics. Eq. (10) leads to the following commutation relations for the fields:
(9)
(12)
[a +, (a+) *]=[a#, (a7)*]=0, [a+ (k), ( a j ) * ( k ' ) ] = ~ j ~ ( k - k ' ) .
(13)
The first term in the equation for Qo may be written in the form
QO=
dk
[(a+)*c)-(c)-)*a +]
(14)
If n~ is equal to the number of the original operators then due to the independence of c)-, the operator Q°o acts non-trivially on all the ghost fields ?] (k). The matrix fl'? has an inverse (15)
and one can introduce creation operators (d~+)*= ([3F')k:(d])*
Let us introduce linear dependent fields ~ 7 = ~ f l , c / . If the maximal number of their independent components is n,, then the fields 8~ may be represented as follows:
I dk
(10)
ff{g~k(flf~)k:= 6",
+ E E Z eJ'~-~(k)fl~(k)c'~(k) +h.c. •
Qo=
[a/+b~Tad, a~ +a*b~,k] = 0 ,
(16)
such that [c~-(k), (0+)*(k')]+ =6,j~(k-k').
(17)
The operator QO acts in the space spanned by the vectors generated by the operators (a +)*, (a j ) * , ( c ] - ) *, ( (~ + ) * and satisfies the conditions ( 1 ) - ( 6 ). In this case only one generation of ghosts is needed. This situation is realized in the Yang-Mills theory and in other models with irreducible constraints and closed algebra. If the number of independent operators is less than the original number of ghosts (n~ < M~ ) the operator Qo° does not act on some of the fields c) and to construct the operator satisfying the conditions ( 1 ) - ( 6 ) one needs higher ghost generations. The operator Q°o is constructed in the same way as before, the creation operators ((~-+)* being defined by eq. (16) where (fli-~)k: is a right inverse matrix. The next term in Qo looks as follows:
Volume 217, number 1,2
(c, +Cd bdi)C~ \~=
PHYSICS LETTERS B
+h.c.
dk.
(18)
=
(]
Qo= f d k
1
The operator Q~ annihilates all the vectors created by ( a + ) *, ( a 7 ) * , (c~-)*, (?)+)* due to condition (It). The operators ~)+
19 January 1989
- , _= yi/Cj-~ + b,2 d .Cd
(19)
are independent and one may introduce creation operators
Y. [ d * ( k ) c , ~ ( k ) + c * ( k ) d , ( k ) ] ,
O~=1
(26)
where N is the total number of different types of partides. In particular N may be equal to oo. An arbitrary state vector may be represented as a sum of the vectors
IO ) = [ O ) p ~ ® l ~ )
,
(27)
where I¢)ph is a positive norm state, and the vector Ig~) looks as follows:
(~,!)*=d*(Y-')~,, I ~)=
[() + (k), ( ( ) - ) * ( k ' ) ]+ = 8 0 ~ ( k - k ' ) .
E
--10~l'l~'
]""
OL\Pn,,]
O!'Lll ) ' ' "
.(qm.)
(20) * " )...c~(sk.) * " ( *. ( t ", )... ( *4 ( /t ". ) xc~(s,
Defining in the same way (c}+). = (2. (fl~-,)/~,
tll...ll,Pv',llll...t~lN ...... S l ...S ~,,, t ', ...t /.,, nx, q I ...q N × Gk,...k,,../,..Z~ (P l1...p JV x)
(21)
Xdp', ...dt~. ,
(28)
we get the following representation for the Q~:
Q~= fdk(~=l(~]+)*c~-+(c2-)*~)+ ) .
(22)
This operator acts in the space spanned by the operators (~)+)*, ( ~ - ) * , ( c ~ - ) * , ((~+)*. Again if n2 is equal to the number of the original operators c~ then the series in (9) terminates, otherwise we must consider the next term, etc. Therefore the general structure of the operator Qo is given by the following expression:
where the functions G are symmetric with respect to the arguments p,", q~ for fixed c~, and antisymmetric with respect to s~, t~. Substituting the vector (2 7 ) into eq. ( 1 ) we get the following relation between the functions G: [ G")--"~'"n'~,m~-"mB--'m,xt~ I /3
k) ...klJ...k~,Ib..I/j...l,v
N q lt ...q N,,,x, S ~I ...S kx,
dk
t
~ (a~-)*c]-+ I
~=1
(g]+)*c]-
~ G n l . . . n B + I...tlA,ml...m/3+ l...m~¢ [ ~ fl-- 1 kl...k B -- 1...k:VJl .../fl-- 1...IN ~1j I " ' P n/~- t tfl~fll/ / 1 ""//nN,~N
I ~l~fl l a l t l l . . . t l~mNN ,
t l ....+#,_,-2 f l - l ~fl ....+/xx ) = 0 G ll [...O...pl.N.ill [...lVll%
+ ,=~ (~,~+)*c3- + . . . + h . c . ) ,
(23)
where each term operates in a separate subspace. Now we can study the problem of positivity of the physical states norm. It is convenient to change notations. In the following we shall denote all Bose operators as a~, d,, [ a . ( k ) , d*~(k') ] = 6 . a 8 ( k - k '
),
(24)
and all Fermi operators as c., c~., [c,~(k), e~(k') ] + = ~ a ~ ( k - k ' ) .
t I ... t )x) A,~
+ (m/~+ 1 )( -- 1 ):;~_, k,+~_~/.
q,1 . . . ,n, i, l, -
Qo=
(25)
In these notations the operator Qo looks as follows:
~/~ I¢
~II I " " P n x )
~,.k,,/~_lx
s lI...S~k[]_ It s2/3...S~, ,
(m,,l/~0)
[...Itl~ =Gk,]l [...]~1%'.t?l o ~x.I, Iu=O.
, (29)
In this equation it is understood that the functions G are symmetrized according to the corresponding Young tableaux which can be easily deduced from the representation (28). If we now calculate the contribution to the norm of the terms with occupation numbers (n~...na...nN, ml ...m/~...mN, kj ...kl~...kN, Ii ...lfl...lN) and (nl ...na + 1 ...nx, m~ ...rn/~ + l...mx, k~ ...k/~- 1...kN, Ii ...[~-- 1.../N) and take into account relation (29) we find that these terms have the same absolute value but differ by a sign. Therefore they cancel in the sum and only the terms with zero occupation numbers give a non-zero contribution. 93
Volume 217, number 1,2
PHYSICS LETTERS B
An arbitrary state vector satisfying eq. ( 1 ) may be represented in the form
10) : 10)phG 10o),
(30)
where I0 ) ph has a positive n o r m a n d I0o) has a zero norm. The vectors 10o) are orthogonal to all other vectors. Hence we may identify all the vectors differing by ]~o). The positive n o r m physical space is the subspace ( 1 ) factorized with respect to the null vector space ~o. So we proved that if the theory satisfies the conditions ( 1 ) - (6) the S-matrix is unitary in the positive n o r m subspace and physical unitarity holds. No assumptions about the gauge group were needed. The only i m p o r t a n t limitation is the hypothesis of asymptotic dynamics linearization. It is valid in standard
94
19 January 1989
field theory at least in the framework of perturbation theory. However, in some models like first quantized strings it may not be true. The conditions ( 1 ) - ( 6 ) are met in a natural way in the BRST approach to gauge theories. Some examples of this procedure a n d a more detailed discussion will be given elsewhere.
References [ 1] T. Kugo and I. Ogima, Suppl. Prog. Theor. Phys. 66 ( 1979 ) 1. [2] G. Curgi and R. Ferrary, Nuovo Cimento 35A (1976) 273. [ 3 ] I.A. Batalinand G. Vilkovisky,Phys. Rev. D 28 ( 1983) 2567. [4 ] I.A. Batalin and E.S. Fradkin, Phys. Lett. B 122 ( 1983 ) 157. [5] M. I-]enneaux,Phys. Rep. 126 (1985) 1. [6] R. Marnelius, Phys. Lett. B 99 ( 1981 ) 467. [ 7 ] L.D. Faddeevand A.A.Slavnov,Gaugetheories; introduction to quantum theory (Benjamin, New York, 1980).