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Hamiltonian brst formalism for gauge-invariant systems with closed algebra
Nuclear Physics B365 (1991) 335-353 North-Holland
HAMILTONIAN
NUCLEAR PHYSICS
B
BRST FORMALISM FOR GAUGE-INVARIANT SYSTEMS WITH CLOSED ALGEBRA G.N. RYBKIN
Institute
for Nuclear Research Anniversary
of the Academy of Sciences of the USSR, 60th October Prospect, 7a, Moscow 117312, USSR
Received 14 May 1991
The systems for which the algebra of gauge transformations in the lagrangian formalism is closed, are considered. The hamiltonian BRST charge and the BRST-invariant hamiltonian are found explicitly. Their expansions in powers of the ghost variables contain, in general, an infinite number of terms.
1. Introduction Usually an action or a lagrangian is used to specify a physical system. For any gauge-invariant system the lagrangian is, by necessity, singular. As a result, in the hamiltonian formalism constraints arise. The main method to construct a covariant quantum theory of such systems is the BRST quantization method. This method is furthest developed in its hamiltonian form, which is called the Batalin-FradkinVilkovisky (BFV) approach [ 1,2]. As is known, the BRST quantization method is based on the principle of BRST symmetry. This symmetry was discovered as invariance of the effective action for the Yang-Mills field under global nilpotent transformations with one odd parameter [3,4]. In the BFV approach the corresponding transformations are generated by the hamiltonian BRST charge, which is determined by the constraints. The effective hamiltonian is defined by means of adding BRST-invariant terms of a special form to some minimal BRST-invariant hamiltonian. It is proved that the hamiltonian BRST charge and the corresponding BRST-invariant hamiltonian exist [5,6]. Explicit expressions for these quantities are found for many physically interesting gauge-invariant systems (see e.g. ref. [5], and references therein). However, this problem is not solved for the general case. In the present paper we construct the hamiltonian BRST charge and the BRST-invariant hamiltonian for gauge-invariant systems for which the algebra of gauge transformations in the lagrangian formalism is closed. This problem is not 0550-3213/91/$03.50
0 1991 - Elsevier Science Publishers B.V. All rights reserved
336
G. N. Rybkin
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Gauge-invariant
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trivial, since in the hamiltonian formalism the gauge algebra generated by the constraints is, in general, open for such systems. Consequently, nonzero higherorder structure functions arise. This paper is organized as follows. In sect. 2 we consider gauge-invariant lagrangian systems with closed algebra. In sects. 3 and 4 the hamiltonian description of such systems is given and first-order structure functions of the constraint algebra are investigated. Explicit expressions for the hamiltonian BRST charge and the BRST-invariant hamiltonian are found in sect. 5. The results obtained and possible generalizations are discussed in sect. 6. All second-order structure functions for systems under consideration are calculated in appendix A. Throughout the paper we assume the summation to be over repeated indices.
2. Gauge algebra Consider a mechanical system, given by a lagrangian L(q, 4>, where q and 4 stand for a set of generalized coordinates q’ and a set of generalized velocities $, r=l , , . . , N. Let us present the Euler-Lagrange equations in the form
where
The matrix W,, is called the hessian. Suppose that for arbitrary infinitesimal functions of time I,, LY= 1,. . . , M, the transformations of a trajectory
are symmetry transformations
for the lagrangian
Uq,
4):
6,L = -$. From eq. (2.2) it folloivs [7] that the function -C, has the form
(2.2)
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and, besides, that the following equalities are valid: *:w,,=o, 5xs
+ -$(Wr)
tPw&~R,)
(2.4)
= 0,
(2.5)
=o.
(2.6)
These equalities are consequences of the Noether identities. From eq. (2.4) it follows that the lagrangian of the system is singular, and $L(q) are null vectors of the hessian W,,(q, 4). We shall assume that these vectors are linearly independent, and that W,, has no other null vectors which are linearly independent from $L. We shall also assume that the algebra of gauge transformations (2.1) is closed: [L#f(f)
=Qf(t).
(2.7)
Substituting eq. (2.1) into eq. (2.7) and taking into account the linear independence of the vectors I& we obtain the following relations [8]:
P-9 P-9) (2.10)
(2.11)
(2.12) Here A&&q, 41, B&(q) are the structure functions of the algebra of gauge transformations. They obey the equalities AYd = -AY Pa ’
(2.13)
338
G.N.
Rybkin
and give the correspondence cp”(t, q(t), 4(t)) in eq. (2.71,
/
Gauge-invariant
between
the
systems
gauge parameters
.P(t), p”(t),
pa = &qPA;y+ (e”py - pW)B,q..
(2.14)
From the Jacobi identity [S,,[6,,6,]]q’(f)
(~“(t>,~“(t>,@(t))
+c~cl.pem
=O
(2.15)
we get B:yB;6-B;yB~~-~;aqr=0,
aA’,,
(2.16)
(2.18)
+ljr-
3. Primary Now we proceed lagrangian L. Let define the mapping help of the familiar
a4rp B$ + cycl. perm.( cx,p, r) = 0. ad
(2.19)
constraint surface and standard extension of functions to the hamiltonian description of our system, given by the us consider the phase space with the coordinates q’,p, and from the state space of the system to the phase space with the reIation P,(4,4)
=
a-w 4) agr *
(3.1)
From eq. (3.1) with eq. (2.4) taken into account it follows that for fixed values of the generalized coordinates q’ and velocities 4’ all the points of the M-dimen-
G.N. Rybkin
sional surface having the parametric
/
Gauge-invariant
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representation
339
of the form
4’(A) =4’, 4’(A) =4’+A”$:(4)
>
(3.2)
are mapped into one and the same point of the phase space. It is easy to understand that the image of the state space is in our case the (2N -M)dimensional surface in the phase space which may be given by M functionally independent conditions: @Jq,p)=O,
ff=l,...,
M.
(3.3)
The functions @, are the so-called primary constraints [9]. It is natural to call the surface in the phase space, given by eq. (3.3), the primary constraint surface [8]. Differentiating the identity
with respect to 4’, we obtain
W,,(4,4)~(4,P(O?Q)) s
=o.
(3.5)
From eqs. (3.41, (3.5) and (2.4) it follows that we can take the functions
(3.6) where U! is some invertible matrix, as the primary constraints. Let a function f(q, 4) be constant on the surfaces in the state space of the system given by eq. (3.21, then the following equalities are valid:
Suppose that each point of the primary constraint surface is an image of only one connected surface of the form (3.2). Then one can find a function F(q,p) such that
Equality (3.8) specifies the values of the function
F(q,
p) only at the points of the
340
G.N. Rybkin / Gauge-invariant systems
primary constraint surface. Then the function F may be extended arbitrarily from the primary constraint surface to the whole phase space. In this, various extensions differ from each other by linear combinations of the primary constraints Ga. Working in the hamiltonian formalism, we must calculate the Poisson brackets of some functions (in particular, hamiltonian and constraints) originally defined on the primary constraint surface. Therefore it is necessary to fix the extension of the functions from the primary constraint surface. We shall consider the so-called standard extension of functions [8, lo]. Let us introduce the vectors x:(q) which are dual to the vectors u:(q)@-;;(q):
To each point of the phase space with the coordinates q’, p, we let correspond the point of the phase space with the coordinates q’,pF, where P,o(cL P) =p,
- @a(43 p)xra(q)
*
(3.10)
In view of eq. (3.6), it is easy to see that @,(q,
Therefore the point function function
P0(4Y PI)
= O.
(3.11)
the point with coordinates qr,pF may be considered as a projection of with coordinates q’,p, onto the primary constraint surface. For any F(q,p), specified on the primary constraint surface at least, we define a F’(q,p) in the whole phase space by the relation
F”(a p) = F(q, PO).
(3.12)
We shall call the function F” the standard extension of the function F from the primary constraint surface, or simply the standard function which coincides with F on the primary constraint surface. Standard extension of functions has a clear geometrical interpretation. For any point of the phase space the value of the function F” is equal to the value of the function F at the projection of this point onto the primary constraint surface. Let us present the main properties of the standard extensions. Obviously, (F+G)‘=F’+G’, (F-G)‘=F”*Go.
(3.13a) (3.13b)
Using (3.9), one can easily get convinced that for partial derivatives of the primary
G.N. Rybkin
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341
constraints (3.6) the following equalities are valid:
J@a a@cl O apr- i-iap, 1
(3.14a)
adja aQa O a@, ax: @ aqr - i-1aqr --- ap, aqr E'
(3.14b)
With the help of eqs. (3.10), (3.14b) and (3.6), we obtain analogous equalities partial derivatives of the standard function,
aF" -=
aF" O -
ap,
i a~, 1 )
wo -= aqr
aF" O aF"ag --@ ( aq* 1 ap, aqr Es
for
(3.15a)
(3.15b)
Taking into account (3.13) and (3.15), it is easy to see that for standard extensions F” and Go the equality
(FO,GO) ={F”,Go)o+~($-$)~@~
(3.16)
is valid [8]. Here and henceforth we consider that nonzero Poisson brackets for the variables q’, p, have the form IPr,qS}
=
--a;.
Equality (3.16) is also valid in the case when either of the functions F” and Go, or both, are primary constraints. It follows immediately from eq. (3.14). Suppose that one can find such functions $Yq), so that the matrix $~aqp/aq’ be invertible. Then, we can fix the matrix ut in eq. (3.6) by setting (3.17) From eqs. (3.9) and (3.17) we see that the vectors (3.18)
G.N. Rybkin / Gauge-invariant systems
342
may be chosen as dual to the vectors u~t,$. Then eq. (3.16) has the most simple form, (F”,Go}
(3.19)
= [F”,Go}‘.
Finally, we list some useful relations. Let W”(q, 4) be the pseudo-inverse matrix to the hessian [ll], defined by the relations
wr,wts= liy )
which is uniquely
x,”W”’= 0.
(3.20),(3.21)
Here we use the notation
Suppose that eqs. (3.7) and (3.8) hold. Then, acting on the identity
FO(qsP(qs 4)) =f(4,4) by the operator get
W’“a/&j”
time and again and using eqs. (3.10),(3.20)-(3.221,
~“~“(w-+~~)) =w’,s, a W’IPN aqsl *-*
we
(3.24)
ap,, . ** Qr,
where n=0,1,2 ,... . The relations (3.24) may be used as the definition of the function F’(q,p) [8]. Differentiating the identity (3.23) with respect to q’ and using eq. (3.24), we find
$LP(4.4))
=
(3.25)
4. Algebra of constraints and hamiltonian Let us go back to the hamiltonian description of the system under consideration, In this section and in what follows we use as the primary constraints the functions @= in the form (3.6), where the matrix U! is defined by eq. (3.17). All the other functions are standard extensions of some functions originally defined on the primary constraint surface. For brevity, the superscript “0” for these functions will be omitted.
G.N. Rybkin
We choose as a hamiltonian the condition
/ Gauge-invariant
systems
of our system the standard function
H(q,p(wi))
=P$.
343
H, satisfying
-L.
(4.1)
Using the consequences (2.41, (2.5) of the Noether identities, the explicit form (3.6) of the primary constraints, and eqs. (3.24) and (3.25), we get
{%HJ(q&U))
=4@,Pr).
(4.2)
The standard functions *a, satisfying the conditions
(4.3) are the secondary constraints [9]. From eqs. (3.19) and (4.2) it follows that
{@&,p),H(w)j
=4b)$(w)-
(4.4)
Analogously, using the consequences (2.4)-(2.6) of the Noether identities, conditions (2.8) and (2.10) for the closure of the gauge algebra, and relations (3.24) and (3.25) for the partial derivatives of the standard functions, one can obtain the remaining expressions for the Poisson brackets of the constraints and the hamiltonian [8]: (4.5) V-6) (1) {~&m-&H(w)}=
-H!~&&LP)~
(4.7)
(1) (~&d4,y&,P)} (1)
=
-wg9(4,P)~$LP).
(4.8)
(1)
Here U$,, HE are the standard functions, satisfying the conditions
(4.9 (1)
Ht(q,p(q,d))
= -(qf+d%&
(4.10)
344
G.N. Rybkin / Gauge-inoariant systems
where the functions u$q, 4), $(q, 4) are defined by the relations (4.11)
(4.12) (1) . . The quantltres ‘;: Is, Ht together with u~B,&,u~ are called the first-order structure functions of the algebra of the constraints and the hamiltonian [5]. Let us investigate the form of the first-order structure functions. From eq. (2.8), with the help of (3.17), we get (4.13) Since
we may introduce the standard functions [$(q, p), which satisfy the conditions (4.14) Now, from eq. (4.13), using the properties (3.13) of the standard extensions, the explicit form (3.6) of the primary constraints @,, we find for u:BBy, the following representation: (4.15) From eqs. (4.10)-(4.14) extensions, one obtains
and the properties
(3.13), (3.15a), (3.24) of the standard
(4.16) where II: are the standard functions, which satisfy the conditions (4.17)
G.N. Rybkin / Gauge-invariant systems
345
and possess the property
(I _ aHaes ahp
Analogously,
(4.18)
ap, apr a
ap,
from eqs. (4.91, (4.11&4.14),
(2.5) and (4.31, it follows that
(1)
KwL
PI = i
!$;A?, r
P)
+ q&l,
P) 3
(4.19)
IA
where A means completely antisymmetric with respect to the lower-indices and w$ are the standard functions, satisfying the conditions
part,
(4.20)
Taking into account eqs. (2.11) and (4.13), and the properties (3.13>,(3.15a>,(3.24) of the standard extensions, it is not difficult to show that
A’ 5. Hamiltonian
(4.21)
BRST (BF’V) formalism
At the present stage we have considered the hamiltonian system with the constraints CD,,Fa and the hamiltonian H. From eqs. (4.4H4.8) it follows that the constraints Qa, qa form a set of first-class constraints, and the hamiltonian H is a first-class quantity [93. We consider the constraints Qa, ?J’a to be irreducible. In order to construct the hamiltonian BRST (BFV) formalism [1,2,51 we enlarge the phase space of the system by adding to the initial (even) variables qr,pr odd variables sd”, 8*, corresponding to the primary and secondary constraints, and variables ga, p, canonically conjugate to them. Thus, the nonzero Poisson brackets for the new, so-called ghost, variables have the form
Let us define a set of functions equations
fir(q, p, %Y’, 3)
6,=P,-8Y,q(q,lw$,
as a solution of the system of
(5-l)
346
G.N. Rybkin
/ Gauge-invariant
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where the functions [tr are specified with the help of eq. (4.14). This system has a unique solution, which may be presented as a set of, in general, infinite series in powers of the ghost variables. Using definition (5.11, one can derive the following relations:
{firdls}= --a;-- Y$-,d) 2 I
(5.2b)
{-%A} =s,q@p - {E,B,}> > ,
(5.2c)
{i&r} =mps- Jg$,P.) > r
(5.2d)
where the notation (5.3) is introduced. Define the function
d(q,
p,
d=8”lv,(q,$)
$7, @) by the equality +SPvau;p(q,fi)97,
where the functions wY,a are determined (5.21, (4.21) only, we get
with the help of eq. (4.20). Using eqs.
{.ii,.ii) = -w%3” {Ya,YB} + 2~~$Py)(q,jj) (
(5.4)
G.N. Rybkin
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347
where the functions ‘;: $ are given by eq. (4.19). From eq. (4.8) it follows that the first term on the right-hand side of eq. (5.5) equals zero. With the help of identity (&14) from appendix A, we find that the second term on the right-hand side of eq. (5.5) is equa1 to zero too. Thus, we have
{f2,Li}=O. Define the function
I-&q,
p, ‘27, s”>,
(5.6)
assuming that
where the functions hz are specified by (4.17). Using eqs. (5.2),(4.18),(4.21) one can convince oneself that
(1)
only,
(1)
where the functions Ht, U& are given by the expressions (4.16) and (4.19). Now, with the help of eq. (4.7) and identity (A.241 from appendix A, we find that
{,ii,ri) =o.
(5.9
Now, the only thing which remains to be done is to include the primary constraints into our construction. Let us consider the functions O,Z, defined in the enlarged phase space by the equalities x = j-j -pd”usjz a P’
L?=i-zi;i+9”@&,fi),
(5.10), (5.11)
With the help of relations (5.21, (4.15), (4.191, (4.21) and, besides, equalities (5.61, (4.5), (4.6), (A.25) and (A.26), we see that {,n,il)
=o.
(5.12)
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Using relations (5.2), (4.15), (4.16), (4.18), (4.19), (4.21) and taking equalities (5.91, (4.4), (A.27) and (A.28), we get (L&x)
= 0.
into
account (5.13)
Thus, 0 is the hamiltonian BRST charge and GY is BRST-invariant hamiltonian. The general form of the BRST-invariant hamiltonian is given by the formula
H,=S-{~,Y},
(5.14)
where ?P is an arbitrary odd function.
6. Conclusion In the present paper we have considered gauge-invariant lagrangian systems for which the algebra of gauge transformations (2.1) is closed. After investigating the algebra of the constraints, which arise in the hamiltonian formalism for such systems, we have found explicit expressions for the hamiltonian BRST charge R and the BRST-invariant hamiltonian Z. In this, the functions $,, defined in the enlarged phase space by the relations (5.0, play a key role. Expanding the functions 5, as a series in powers of the ghost variables one may obtain the corresponding expansions for R (5.4),(5.10) and Z (5.7),(5.11). Ail structure functions of the theory appear as the coefficients of these expansions [2,5]. In particular,
The corresponding series for fi (5.4) contains, in general, an infinite terms. Retaining three lowest-order terms we get
number of
where the notation (5.3) is used. From eq. (6.3) it is easy to see that first- and second-order structure functions are given by equalities (4.19) and (A.12). From {6,6} = 0 it follows that these structure functions obey eqs. (4.8) and (A.4).
G.N. Rybkin
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349
Equalities of such type arise for the higher-order structure functions too. They may be considered as the definition of the structure functions [2,5]. Inthis paper we have assumed that the initial system is described with the help of even variables. It is not difficult to generalize our consideration to systems described by both even and odd variables [7,12]. In a similar way one may consider the case when the quantities I,&: depend not only on the generalized coordinates but also on the generalized velocities, like the quantities 5:. The author is indebted work.
to A.V. Razumov who has suggested the theme of this
Appendix A SECOND-ORDER
STRUCTURE
FUNCTIONS
The Jacobi identity (PY~{lyn~lvg~j)*=o~ where A means completely antisymmetric help of (4.8) be rewritten in the form
(A4
in the lower-indices part, may with the
(1)
( 1 DGY
Y,=O.
A
(A.4
Here we have denoted I.51
(A.31 In virtue of eq. (A.21 and the theorem proved in ref. [5], there exist functions such that
(1) =2US" Y ( 1A eJ-Y2:'
Us,,
(2)
DtY
(2) u$jy=
-
(2) u;&.
(2)
The quantities
Us,, are called the second-order structure functions.
(A4
(A.5)
350
G.N. Rybkin
/ Gauge-inuariant
Let us calculate the functions
systems
on the primary constraint surface. For
this purpose it is convenient to use the
.,,~WSf-$
+4’$-$
+p-, af
(A-6)
yaqS
where F is the standard function, satisfying condition (3.23). One can derive formula (A.6) with the help of the relations (4.3>,(3.7),(3.24),(3.25) and the consequences (2.4)-(2.6) of the Noether identities. Next, using the equalities (3.17), (4.9), (4X), the conditions (2.9), (2.10), (2.12) for the closure of the gauge algebra, the consequences (2.17)-(2.19) of the Jacobi identity (2.15), we obtain
(~bey(a,P(%l)))A= -(
.
(A.7)
A
Taking into account eqs. (2.8), (2.11) and (2.5), it is easy to show that
(A-9) where I7,S is given by eq. (3.22). With the help of (A.81, (A.9) and (3.20) we can rewrite (A.7) as
) (tk%). (A.10) A
From here, allowing
for properties
extensions and equalities by
(3.13), (3.15a), (3.19), (3.24) of the standard (2)
(4.3), (4.141, we come to relation (A.4) with U &, given
(A.ll)
G.N. Rybkin / Gauge-invariant systems
Using eqs. (4.191, (4.21), we immediately
351
obtain
(A.12) By virtue of eq. (A.12) it is obvious that equalities (A.5) are valid. Acting on (4.8) by the operator [t;“r a/ap,., we get (2)
=-2US”
a&pE’
(A.13)
A
where equalities (A.111 are used. Then, with the help of eqs. (A.31, (A.41, (A.13) and (4.191, we conclude that
(1)
(1)
+2u;,u;,-
(1)
aPE
u:p-+;r
(A.14)
r This identity is applied to prove (5.6) in the main text.
(2)
Analogously, the second-order structure functions HzY for the hamiltonian be found. Taking into account eqs. (4.7) and (4.8), the Jacobi identity
can be rewritten as (A.16) where we have set [5]
Next, one can apply (A.6) together with the formula (H,F}(q,p(q,cj))
=f.f
($:+*:)-$ (
+ec$)
-w+$
-4’S
(Aw
G.N. Rybkin / Gauge-inuariant systems
352
A’
This results in (2) A
(A.19)
= 2HS”Lyp‘I’ C)
where (A.20)
With
the help of eqs. (A.201, (4.16) and (4.1% we see that (2) (2) HFD = - HE6e,
(A.21) (A.22)
From eqs. (4.7), (A.21) and (A.221 it follows that
(A.23)
Equalities
(A.231 together with (A.19) give
(1) (1)
+2H;
(1) alu,
U,“, - HE +;,.r
(1)
U&H:
(1)
(A.24)
where we used eqs. (A.17), (4.16), (4.19). Identity (A.241 is used to prove (5.9) in the main text. In a similar way one may consider the remaining Jacobi identities for the Poisson brackets of the functions @=, T,, H and perform corresponding calcula-
G. N. Rybkin
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Gauge-inuariant
353
systems
tions. These result in
((~a,U~B:s}+u~B:,.u~B,r,)-(ol*p)=O,
(A.25)
(A.26) (A.27)
(1)
(1)
(1)
+ H”.u’B’ P a SE-u:BpEs. H:+uf:*2U;,=O.
(A.28)
In other words, the remaining second-order structure functions of the algebra of the constraints and the hamiltonian are equal to zero. References [l] E.S. Fradkin and G.A. Vilkovisky, Phys. Lett. B55 (1975) 224; LA. Batalin and G.A. Viikovisky, Phys. Lett. B69 (1977) 309 [2] ES. Fradkin and T.E. Fradkina, Phys. Lett. B72 (1978) 343; I.A. Batalin and ES. Fradkin, Phys. Lett. B122 (19831 157; Riv. Nuovo Cim. 9, No. 10 (1986) 1 [3] C. Becchi, A. Rouet and R. Stora, Phys. Lett. B52 (1974) 344; Commun. Math. Phys. 42 (1975) 127; Ann. Phys. (N.Y.) 98 (1976) 287 [4] IV. Tyutin, FIAN preprint No. 39 (19751, in Russian [5] M. Henneaux, Phys. Rep. 126 (1985) 1 [6] I.A. Bataiin, P.M. Lavrov and IV. Tyutin, J. Math. Phys. 31 (1990) 6 [7] A.V. Razumov, in Quantum field theory and high energy physics (Moscow State University, Moscow, 19871, in Russian; P.N. Pyatov, A.V. Razumov and G.N. Rybkin, IHEP preprint 88-212 (1988) [8] P.N. Pyatov and A.V. Razumov, Int. J. Mod. Phys. A4 (1989) 3211 [9] P.A.M. Dirac, Lectures on quantum mechanics (Yeshiva University, New York, 1964) [lo] Kh.S. Nirov and A.V. Razumov, IHEP preprint 90-45 (1990) [ll] P. Lankaster, Theory of matrices (Academic Press, New York, 196); A.V. Razumov, IHEP preprint 84-86 (1984) [12] B. Dewitt, Supermanifolds (Cambridge Univ. Press, Cambridge, 1984)
HAMILTONIAN
NUCLEAR PHYSICS
B
BRST FORMALISM FOR GAUGE-INVARIANT SYSTEMS WITH CLOSED ALGEBRA G.N. RYBKIN
Institute
for Nuclear Research Anniversary
of the Academy of Sciences of the USSR, 60th October Prospect, 7a, Moscow 117312, USSR
Received 14 May 1991
The systems for which the algebra of gauge transformations in the lagrangian formalism is closed, are considered. The hamiltonian BRST charge and the BRST-invariant hamiltonian are found explicitly. Their expansions in powers of the ghost variables contain, in general, an infinite number of terms.
1. Introduction Usually an action or a lagrangian is used to specify a physical system. For any gauge-invariant system the lagrangian is, by necessity, singular. As a result, in the hamiltonian formalism constraints arise. The main method to construct a covariant quantum theory of such systems is the BRST quantization method. This method is furthest developed in its hamiltonian form, which is called the Batalin-FradkinVilkovisky (BFV) approach [ 1,2]. As is known, the BRST quantization method is based on the principle of BRST symmetry. This symmetry was discovered as invariance of the effective action for the Yang-Mills field under global nilpotent transformations with one odd parameter [3,4]. In the BFV approach the corresponding transformations are generated by the hamiltonian BRST charge, which is determined by the constraints. The effective hamiltonian is defined by means of adding BRST-invariant terms of a special form to some minimal BRST-invariant hamiltonian. It is proved that the hamiltonian BRST charge and the corresponding BRST-invariant hamiltonian exist [5,6]. Explicit expressions for these quantities are found for many physically interesting gauge-invariant systems (see e.g. ref. [5], and references therein). However, this problem is not solved for the general case. In the present paper we construct the hamiltonian BRST charge and the BRST-invariant hamiltonian for gauge-invariant systems for which the algebra of gauge transformations in the lagrangian formalism is closed. This problem is not 0550-3213/91/$03.50
0 1991 - Elsevier Science Publishers B.V. All rights reserved
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trivial, since in the hamiltonian formalism the gauge algebra generated by the constraints is, in general, open for such systems. Consequently, nonzero higherorder structure functions arise. This paper is organized as follows. In sect. 2 we consider gauge-invariant lagrangian systems with closed algebra. In sects. 3 and 4 the hamiltonian description of such systems is given and first-order structure functions of the constraint algebra are investigated. Explicit expressions for the hamiltonian BRST charge and the BRST-invariant hamiltonian are found in sect. 5. The results obtained and possible generalizations are discussed in sect. 6. All second-order structure functions for systems under consideration are calculated in appendix A. Throughout the paper we assume the summation to be over repeated indices.
2. Gauge algebra Consider a mechanical system, given by a lagrangian L(q, 4>, where q and 4 stand for a set of generalized coordinates q’ and a set of generalized velocities $, r=l , , . . , N. Let us present the Euler-Lagrange equations in the form
where
The matrix W,, is called the hessian. Suppose that for arbitrary infinitesimal functions of time I,, LY= 1,. . . , M, the transformations of a trajectory
are symmetry transformations
for the lagrangian
Uq,
4):
6,L = -$. From eq. (2.2) it folloivs [7] that the function -C, has the form
(2.2)
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and, besides, that the following equalities are valid: *:w,,=o, 5xs
+ -$(Wr)
tPw&~R,)
(2.4)
= 0,
(2.5)
=o.
(2.6)
These equalities are consequences of the Noether identities. From eq. (2.4) it follows that the lagrangian of the system is singular, and $L(q) are null vectors of the hessian W,,(q, 4). We shall assume that these vectors are linearly independent, and that W,, has no other null vectors which are linearly independent from $L. We shall also assume that the algebra of gauge transformations (2.1) is closed: [L#f(f)
=Qf(t).
(2.7)
Substituting eq. (2.1) into eq. (2.7) and taking into account the linear independence of the vectors I& we obtain the following relations [8]:
P-9 P-9) (2.10)
(2.11)
(2.12) Here A&&q, 41, B&(q) are the structure functions of the algebra of gauge transformations. They obey the equalities AYd = -AY Pa ’
(2.13)
338
G.N.
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and give the correspondence cp”(t, q(t), 4(t)) in eq. (2.71,
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the
systems
gauge parameters
.P(t), p”(t),
pa = &qPA;y+ (e”py - pW)B,q..
(2.14)
From the Jacobi identity [S,,[6,,6,]]q’(f)
(~“(t>,~“(t>,@(t))
+c~cl.pem
=O
(2.15)
we get B:yB;6-B;yB~~-~;aqr=0,
aA’,,
(2.16)
(2.18)
+ljr-
3. Primary Now we proceed lagrangian L. Let define the mapping help of the familiar
a4rp B$ + cycl. perm.( cx,p, r) = 0. ad
(2.19)
constraint surface and standard extension of functions to the hamiltonian description of our system, given by the us consider the phase space with the coordinates q’,p, and from the state space of the system to the phase space with the reIation P,(4,4)
=
a-w 4) agr *
(3.1)
From eq. (3.1) with eq. (2.4) taken into account it follows that for fixed values of the generalized coordinates q’ and velocities 4’ all the points of the M-dimen-
G.N. Rybkin
sional surface having the parametric
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representation
339
of the form
4’(A) =4’, 4’(A) =4’+A”$:(4)
>
(3.2)
are mapped into one and the same point of the phase space. It is easy to understand that the image of the state space is in our case the (2N -M)dimensional surface in the phase space which may be given by M functionally independent conditions: @Jq,p)=O,
ff=l,...,
M.
(3.3)
The functions @, are the so-called primary constraints [9]. It is natural to call the surface in the phase space, given by eq. (3.3), the primary constraint surface [8]. Differentiating the identity
with respect to 4’, we obtain
W,,(4,4)~(4,P(O?Q)) s
=o.
(3.5)
From eqs. (3.41, (3.5) and (2.4) it follows that we can take the functions
(3.6) where U! is some invertible matrix, as the primary constraints. Let a function f(q, 4) be constant on the surfaces in the state space of the system given by eq. (3.21, then the following equalities are valid:
Suppose that each point of the primary constraint surface is an image of only one connected surface of the form (3.2). Then one can find a function F(q,p) such that
Equality (3.8) specifies the values of the function
F(q,
p) only at the points of the
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G.N. Rybkin / Gauge-invariant systems
primary constraint surface. Then the function F may be extended arbitrarily from the primary constraint surface to the whole phase space. In this, various extensions differ from each other by linear combinations of the primary constraints Ga. Working in the hamiltonian formalism, we must calculate the Poisson brackets of some functions (in particular, hamiltonian and constraints) originally defined on the primary constraint surface. Therefore it is necessary to fix the extension of the functions from the primary constraint surface. We shall consider the so-called standard extension of functions [8, lo]. Let us introduce the vectors x:(q) which are dual to the vectors u:(q)@-;;(q):
To each point of the phase space with the coordinates q’, p, we let correspond the point of the phase space with the coordinates q’,pF, where P,o(cL P) =p,
- @a(43 p)xra(q)
*
(3.10)
In view of eq. (3.6), it is easy to see that @,(q,
Therefore the point function function
P0(4Y PI)
= O.
(3.11)
the point with coordinates qr,pF may be considered as a projection of with coordinates q’,p, onto the primary constraint surface. For any F(q,p), specified on the primary constraint surface at least, we define a F’(q,p) in the whole phase space by the relation
F”(a p) = F(q, PO).
(3.12)
We shall call the function F” the standard extension of the function F from the primary constraint surface, or simply the standard function which coincides with F on the primary constraint surface. Standard extension of functions has a clear geometrical interpretation. For any point of the phase space the value of the function F” is equal to the value of the function F at the projection of this point onto the primary constraint surface. Let us present the main properties of the standard extensions. Obviously, (F+G)‘=F’+G’, (F-G)‘=F”*Go.
(3.13a) (3.13b)
Using (3.9), one can easily get convinced that for partial derivatives of the primary
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constraints (3.6) the following equalities are valid:
J@a a@cl O apr- i-iap, 1
(3.14a)
adja aQa O a@, ax: @ aqr - i-1aqr --- ap, aqr E'
(3.14b)
With the help of eqs. (3.10), (3.14b) and (3.6), we obtain analogous equalities partial derivatives of the standard function,
aF" -=
aF" O -
ap,
i a~, 1 )
wo -= aqr
aF" O aF"ag --@ ( aq* 1 ap, aqr Es
for
(3.15a)
(3.15b)
Taking into account (3.13) and (3.15), it is easy to see that for standard extensions F” and Go the equality
(FO,GO) ={F”,Go)o+~($-$)~@~
(3.16)
is valid [8]. Here and henceforth we consider that nonzero Poisson brackets for the variables q’, p, have the form IPr,qS}
=
--a;.
Equality (3.16) is also valid in the case when either of the functions F” and Go, or both, are primary constraints. It follows immediately from eq. (3.14). Suppose that one can find such functions $Yq), so that the matrix $~aqp/aq’ be invertible. Then, we can fix the matrix ut in eq. (3.6) by setting (3.17) From eqs. (3.9) and (3.17) we see that the vectors (3.18)
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342
may be chosen as dual to the vectors u~t,$. Then eq. (3.16) has the most simple form, (F”,Go}
(3.19)
= [F”,Go}‘.
Finally, we list some useful relations. Let W”(q, 4) be the pseudo-inverse matrix to the hessian [ll], defined by the relations
wr,wts= liy )
which is uniquely
x,”W”’= 0.
(3.20),(3.21)
Here we use the notation
Suppose that eqs. (3.7) and (3.8) hold. Then, acting on the identity
FO(qsP(qs 4)) =f(4,4) by the operator get
W’“a/&j”
time and again and using eqs. (3.10),(3.20)-(3.221,
~“~“(w-+~~)) =w’,s, a W’IPN aqsl *-*
we
(3.24)
ap,, . ** Qr,
where n=0,1,2 ,... . The relations (3.24) may be used as the definition of the function F’(q,p) [8]. Differentiating the identity (3.23) with respect to q’ and using eq. (3.24), we find
$LP(4.4))
=
(3.25)
4. Algebra of constraints and hamiltonian Let us go back to the hamiltonian description of the system under consideration, In this section and in what follows we use as the primary constraints the functions @= in the form (3.6), where the matrix U! is defined by eq. (3.17). All the other functions are standard extensions of some functions originally defined on the primary constraint surface. For brevity, the superscript “0” for these functions will be omitted.
G.N. Rybkin
We choose as a hamiltonian the condition
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of our system the standard function
H(q,p(wi))
=P$.
343
H, satisfying
-L.
(4.1)
Using the consequences (2.41, (2.5) of the Noether identities, the explicit form (3.6) of the primary constraints, and eqs. (3.24) and (3.25), we get
{%HJ(q&U))
=4@,Pr).
(4.2)
The standard functions *a, satisfying the conditions
(4.3) are the secondary constraints [9]. From eqs. (3.19) and (4.2) it follows that
{@&,p),H(w)j
=4b)$(w)-
(4.4)
Analogously, using the consequences (2.4)-(2.6) of the Noether identities, conditions (2.8) and (2.10) for the closure of the gauge algebra, and relations (3.24) and (3.25) for the partial derivatives of the standard functions, one can obtain the remaining expressions for the Poisson brackets of the constraints and the hamiltonian [8]: (4.5) V-6) (1) {~&m-&H(w)}=
-H!~&&LP)~
(4.7)
(1) (~&d4,y&,P)} (1)
=
-wg9(4,P)~$LP).
(4.8)
(1)
Here U$,, HE are the standard functions, satisfying the conditions
(4.9 (1)
Ht(q,p(q,d))
= -(qf+d%&
(4.10)
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G.N. Rybkin / Gauge-inoariant systems
where the functions u$q, 4), $(q, 4) are defined by the relations (4.11)
(4.12) (1) . . The quantltres ‘;: Is, Ht together with u~B,&,u~ are called the first-order structure functions of the algebra of the constraints and the hamiltonian [5]. Let us investigate the form of the first-order structure functions. From eq. (2.8), with the help of (3.17), we get (4.13) Since
we may introduce the standard functions [$(q, p), which satisfy the conditions (4.14) Now, from eq. (4.13), using the properties (3.13) of the standard extensions, the explicit form (3.6) of the primary constraints @,, we find for u:BBy, the following representation: (4.15) From eqs. (4.10)-(4.14) extensions, one obtains
and the properties
(3.13), (3.15a), (3.24) of the standard
(4.16) where II: are the standard functions, which satisfy the conditions (4.17)
G.N. Rybkin / Gauge-invariant systems
345
and possess the property
(I _ aHaes ahp
Analogously,
(4.18)
ap, apr a
ap,
from eqs. (4.91, (4.11&4.14),
(2.5) and (4.31, it follows that
(1)
KwL
PI = i
!$;A?, r
P)
+ q&l,
P) 3
(4.19)
IA
where A means completely antisymmetric with respect to the lower-indices and w$ are the standard functions, satisfying the conditions
part,
(4.20)
Taking into account eqs. (2.11) and (4.13), and the properties (3.13>,(3.15a>,(3.24) of the standard extensions, it is not difficult to show that
A’ 5. Hamiltonian
(4.21)
BRST (BF’V) formalism
At the present stage we have considered the hamiltonian system with the constraints CD,,Fa and the hamiltonian H. From eqs. (4.4H4.8) it follows that the constraints Qa, qa form a set of first-class constraints, and the hamiltonian H is a first-class quantity [93. We consider the constraints Qa, ?J’a to be irreducible. In order to construct the hamiltonian BRST (BFV) formalism [1,2,51 we enlarge the phase space of the system by adding to the initial (even) variables qr,pr odd variables sd”, 8*, corresponding to the primary and secondary constraints, and variables ga, p, canonically conjugate to them. Thus, the nonzero Poisson brackets for the new, so-called ghost, variables have the form
Let us define a set of functions equations
fir(q, p, %Y’, 3)
6,=P,-8Y,q(q,lw$,
as a solution of the system of
(5-l)
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where the functions [tr are specified with the help of eq. (4.14). This system has a unique solution, which may be presented as a set of, in general, infinite series in powers of the ghost variables. Using definition (5.11, one can derive the following relations:
{firdls}= --a;-- Y$-,d) 2 I
(5.2b)
{-%A} =s,q@p - {E,B,}> > ,
(5.2c)
{i&r} =mps- Jg$,P.) > r
(5.2d)
where the notation (5.3) is introduced. Define the function
d(q,
p,
d=8”lv,(q,$)
$7, @) by the equality +SPvau;p(q,fi)97,
where the functions wY,a are determined (5.21, (4.21) only, we get
with the help of eq. (4.20). Using eqs.
{.ii,.ii) = -w%3” {Ya,YB} + 2~~$Py)(q,jj) (
(5.4)
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347
where the functions ‘;: $ are given by eq. (4.19). From eq. (4.8) it follows that the first term on the right-hand side of eq. (5.5) equals zero. With the help of identity (&14) from appendix A, we find that the second term on the right-hand side of eq. (5.5) is equa1 to zero too. Thus, we have
{f2,Li}=O. Define the function
I-&q,
p, ‘27, s”>,
(5.6)
assuming that
where the functions hz are specified by (4.17). Using eqs. (5.2),(4.18),(4.21) one can convince oneself that
(1)
only,
(1)
where the functions Ht, U& are given by the expressions (4.16) and (4.19). Now, with the help of eq. (4.7) and identity (A.241 from appendix A, we find that
{,ii,ri) =o.
(5.9
Now, the only thing which remains to be done is to include the primary constraints into our construction. Let us consider the functions O,Z, defined in the enlarged phase space by the equalities x = j-j -pd”usjz a P’
L?=i-zi;i+9”@&,fi),
(5.10), (5.11)
With the help of relations (5.21, (4.15), (4.191, (4.21) and, besides, equalities (5.61, (4.5), (4.6), (A.25) and (A.26), we see that {,n,il)
=o.
(5.12)
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Using relations (5.2), (4.15), (4.16), (4.18), (4.19), (4.21) and taking equalities (5.91, (4.4), (A.27) and (A.28), we get (L&x)
= 0.
into
account (5.13)
Thus, 0 is the hamiltonian BRST charge and GY is BRST-invariant hamiltonian. The general form of the BRST-invariant hamiltonian is given by the formula
H,=S-{~,Y},
(5.14)
where ?P is an arbitrary odd function.
6. Conclusion In the present paper we have considered gauge-invariant lagrangian systems for which the algebra of gauge transformations (2.1) is closed. After investigating the algebra of the constraints, which arise in the hamiltonian formalism for such systems, we have found explicit expressions for the hamiltonian BRST charge R and the BRST-invariant hamiltonian Z. In this, the functions $,, defined in the enlarged phase space by the relations (5.0, play a key role. Expanding the functions 5, as a series in powers of the ghost variables one may obtain the corresponding expansions for R (5.4),(5.10) and Z (5.7),(5.11). Ail structure functions of the theory appear as the coefficients of these expansions [2,5]. In particular,
The corresponding series for fi (5.4) contains, in general, an infinite terms. Retaining three lowest-order terms we get
number of
where the notation (5.3) is used. From eq. (6.3) it is easy to see that first- and second-order structure functions are given by equalities (4.19) and (A.12). From {6,6} = 0 it follows that these structure functions obey eqs. (4.8) and (A.4).
G.N. Rybkin
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Equalities of such type arise for the higher-order structure functions too. They may be considered as the definition of the structure functions [2,5]. Inthis paper we have assumed that the initial system is described with the help of even variables. It is not difficult to generalize our consideration to systems described by both even and odd variables [7,12]. In a similar way one may consider the case when the quantities I,&: depend not only on the generalized coordinates but also on the generalized velocities, like the quantities 5:. The author is indebted work.
to A.V. Razumov who has suggested the theme of this
Appendix A SECOND-ORDER
STRUCTURE
FUNCTIONS
The Jacobi identity (PY~{lyn~lvg~j)*=o~ where A means completely antisymmetric help of (4.8) be rewritten in the form
(A4
in the lower-indices part, may with the
(1)
( 1 DGY
Y,=O.
A
(A.4
Here we have denoted I.51
(A.31 In virtue of eq. (A.21 and the theorem proved in ref. [5], there exist functions such that
(1) =2US" Y ( 1A eJ-Y2:'
Us,,
(2)
DtY
(2) u$jy=
-
(2) u;&.
(2)
The quantities
Us,, are called the second-order structure functions.
(A4
(A.5)
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/ Gauge-inuariant
Let us calculate the functions
systems
on the primary constraint surface. For
this purpose it is convenient to use the
.,,~WSf-$
+4’$-$
+p-, af
(A-6)
yaqS
where F is the standard function, satisfying condition (3.23). One can derive formula (A.6) with the help of the relations (4.3>,(3.7),(3.24),(3.25) and the consequences (2.4)-(2.6) of the Noether identities. Next, using the equalities (3.17), (4.9), (4X), the conditions (2.9), (2.10), (2.12) for the closure of the gauge algebra, the consequences (2.17)-(2.19) of the Jacobi identity (2.15), we obtain
(~bey(a,P(%l)))A= -(
.
(A.7)
A
Taking into account eqs. (2.8), (2.11) and (2.5), it is easy to show that
(A-9) where I7,S is given by eq. (3.22). With the help of (A.81, (A.9) and (3.20) we can rewrite (A.7) as
) (tk%). (A.10) A
From here, allowing
for properties
extensions and equalities by
(3.13), (3.15a), (3.19), (3.24) of the standard (2)
(4.3), (4.141, we come to relation (A.4) with U &, given
(A.ll)
G.N. Rybkin / Gauge-invariant systems
Using eqs. (4.191, (4.21), we immediately
351
obtain
(A.12) By virtue of eq. (A.12) it is obvious that equalities (A.5) are valid. Acting on (4.8) by the operator [t;“r a/ap,., we get (2)
=-2US”
a&pE’
(A.13)
A
where equalities (A.111 are used. Then, with the help of eqs. (A.31, (A.41, (A.13) and (4.191, we conclude that
(1)
(1)
+2u;,u;,-
(1)
aPE
u:p-+;r
(A.14)
r This identity is applied to prove (5.6) in the main text.
(2)
Analogously, the second-order structure functions HzY for the hamiltonian be found. Taking into account eqs. (4.7) and (4.8), the Jacobi identity
can be rewritten as (A.16) where we have set [5]
Next, one can apply (A.6) together with the formula (H,F}(q,p(q,cj))
=f.f
($:+*:)-$ (
+ec$)
-w+$
-4’S
(Aw
G.N. Rybkin / Gauge-inuariant systems
352
A’
This results in (2) A
(A.19)
= 2HS”Lyp‘I’ C)
where (A.20)
With
the help of eqs. (A.201, (4.16) and (4.1% we see that (2) (2) HFD = - HE6e,
(A.21) (A.22)
From eqs. (4.7), (A.21) and (A.221 it follows that
(A.23)
Equalities
(A.231 together with (A.19) give
(1) (1)
+2H;
(1) alu,
U,“, - HE +;,.r
(1)
U&H:
(1)
(A.24)
where we used eqs. (A.17), (4.16), (4.19). Identity (A.241 is used to prove (5.9) in the main text. In a similar way one may consider the remaining Jacobi identities for the Poisson brackets of the functions @=, T,, H and perform corresponding calcula-
G. N. Rybkin
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353
systems
tions. These result in
((~a,U~B:s}+u~B:,.u~B,r,)-(ol*p)=O,
(A.25)
(A.26) (A.27)
(1)
(1)
(1)
+ H”.u’B’ P a SE-u:BpEs. H:+uf:*2U;,=O.
(A.28)
In other words, the remaining second-order structure functions of the algebra of the constraints and the hamiltonian are equal to zero. References [l] E.S. Fradkin and G.A. Vilkovisky, Phys. Lett. B55 (1975) 224; LA. Batalin and G.A. Viikovisky, Phys. Lett. B69 (1977) 309 [2] ES. Fradkin and T.E. Fradkina, Phys. Lett. B72 (1978) 343; I.A. Batalin and ES. Fradkin, Phys. Lett. B122 (19831 157; Riv. Nuovo Cim. 9, No. 10 (1986) 1 [3] C. Becchi, A. Rouet and R. Stora, Phys. Lett. B52 (1974) 344; Commun. Math. Phys. 42 (1975) 127; Ann. Phys. (N.Y.) 98 (1976) 287 [4] IV. Tyutin, FIAN preprint No. 39 (19751, in Russian [5] M. Henneaux, Phys. Rep. 126 (1985) 1 [6] I.A. Bataiin, P.M. Lavrov and IV. Tyutin, J. Math. Phys. 31 (1990) 6 [7] A.V. Razumov, in Quantum field theory and high energy physics (Moscow State University, Moscow, 19871, in Russian; P.N. Pyatov, A.V. Razumov and G.N. Rybkin, IHEP preprint 88-212 (1988) [8] P.N. Pyatov and A.V. Razumov, Int. J. Mod. Phys. A4 (1989) 3211 [9] P.A.M. Dirac, Lectures on quantum mechanics (Yeshiva University, New York, 1964) [lo] Kh.S. Nirov and A.V. Razumov, IHEP preprint 90-45 (1990) [ll] P. Lankaster, Theory of matrices (Academic Press, New York, 196); A.V. Razumov, IHEP preprint 84-86 (1984) [12] B. Dewitt, Supermanifolds (Cambridge Univ. Press, Cambridge, 1984)