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Lagrangian and Hamiltonian BRST formalisms
Volume 224, number 3
PHYSICS LETTERS B
29 June 1989
LAGRANGIAN AND HAMILTONIAN BRST FORMALISMS C. BATLLE 1, j. G O M I S 2, j. PARIS and J. R O C A
Departament d'Estructura i Constituents de la Matbria, Universitatde Barcelona, Diagonal647, E-08028Barcelona, Spain Received 14 March 1989
We establish, at the classical level, the equivalence between the standard BRST hamiltonian formalism and the lagrangian BRST approach with fields and antifields for irreducible first rank theories.
Nowadays the BRST symmetry [ 1 ] plays a crucial role in the quantization of gauge theories. At the classical level it constitutes the modern language to study constrained systems [2]. The BRST symmetry can be studied in the hamiltonian [3,4] and lagrangian [ 5 - 8 ] formalisms for irreducible and reducible systems. The aim o f this paper is to study, at the classical level, the relation between both formalisms using the approach with fields and antifields of Batalin and Vilkovisky [ 7 ] as our starting point. We will restrict ourselves to irreducible first rank theories with first class constraints described by an action ~o[¢'2"]. The associated canonical action ~ , is given by
The number o f phase space physical degrees of freedom is 2 ( n - r). The Poisson bracket algebra among T~ and Ho gives
{To,, T/j}=f~/jTy,
{Ho, T,~}= V. PT~,
w h e r e f ~ p and V, p are constant. The canonical action (1) has the gauge transformations 8 ¢ ' = { ¢ i, T.}e",
8P,={pi, To~}e",
82" = ~ " - V/j "eP + f ~,/2~'e/~. The original action ~ [ ~ , 2], with lagrangian
Lo(~, 6, 2), is invariant under 8q~'=FL~{~', T , } e " - a ' , ( ~ ,
~ : = f dt [~'p~-H~°)(O,p, 2)], i=l,...,n,
c e = l ..... r,
H ~°) (0, P, 2) = Ho (0, p) + 2 " T . (¢~,p),
82" = 6" - Vp "e~ + f ~y2 "/ea, (i) (2)
where ~', 2" and p,, n , are the coordinates and their corresponding momenta, respectively ~, and Ho is the "canonical" hamiltonian. The hamiltonian constraints are
7~..~0,
(3)
where FL~ is the pull-back of the Legendre transformation of Lo. In the lagrangian formalism of Batalin and Vilkovisky [ 7] the fields appearing in the classical action 5~oare generally denoted by ~ : ~ =~[~"],
a=l,...,m,
and they are included in a larger set o f " q u a n t u m " fields q~:
T.(O,p)~O.
{Oa} c {~A}, Present address: Department of Physics, Jadwin Hall, University of Princeton, Princeton, NJ 08540, USA. 2 Bitnet QUIM@EBUBECM1. ~ For simplicity we will consider e(Oi)=e(2") =0. The extension to odd variables is trivial. 288
6, 2)e",
A = I ..... M.
To each field ~ A we associate an antifield qb] with opposite statistics: ~(qbA) = ~A, E(qb3) =~A + 1.
0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division)
Volume 224, number 3
PHYSICS LETTERSB
In the space of fields and antifields one can define an odd symplectic structure called antibracket ~2: O~F O~G (F, G) =- Oq~A Oq~
0r F O~G 0~3 Oq)A"
hamiltonian formalism we need to perform the Dirac hamiltonian analysis [2 ] of the action S (5). With this purpose we consider the momenta associated to the fields and antifields,
OL
The "quantum" action S [ ~ , ~ * ] obeys the master equation (S,S)=O.
(4)
29 June 1989
OLo
OL ~ " - 0,~" =0,
OaJ.
p . i _ O~L
O~L lr*~= ~ =0,
The so-called proper solution contains M independent gauge transformations. In this space there is a natural definition of a generalized BRST symmetry [ 9 ]:
From (8) we get the following primary constraints:
a ~ A = (-- 1 ),~,(q)A, S),
~'~0,
a~.~ = ( -- 1 ),A+l ( ~ ,
.~,~÷A* ~ 0,
S).
~=
0~L o~ ~ --~;'
p*i~O,
~,~_
0L - o~---~ =o.
(8)
,~*a~O,
~*~0.
(9) (10)
Since S[@, ~ * ] obeys the master equation (4) the BRST transformation is nilpotent; furthermore, it is a symmetry of S. The proper solution of the master equation, associated to the original action 5eo[0, 2] mentioned above, reads
With the introduction of a graded symplectic structure [ 10] on the phase space of fields and antifields we find that the constraints (10) are second class. Using their associated Dirac bracket [2] we eliminate the canonically conjugate pairs (2*, ~*~). The canonical hamiltonian H is given by
S= f dt[Lo+OTa'~ ~
H = f ' p , - Lo(¢, f, 2) - ¢* a'~ (~, f, 2) ~ + ~ , ( - V~ ~ ~ P + f ~y2y cg~)
+½ ~,, fo~ c g ~ ] .
(5)
This action is invariant under the following BRST transformation: {3Q,=aia ~a, 82~= ~ ~a=
8S 60.__ aQ i'
V~ ~ ' ~ + f ~ r 2 r ~z,, 82" =
t a _~f~
;.
rg~, 8 ~ * - -
~S ~Scg.. .
(6)
+ ~1 2•~ f a~
0 02 ~ (Lo + ¢ * a ~ ~ ) ~'~ (gT~;~
.
(7)
In order to make contact with the standard BRST "~ Summation over discrete and continuous indices is assumed.
~';' ~,~,
where f i are the hamiltonian expression of the velocities 0i obtained through a partial inversion of the Legendre transformation. The stability of the primary constraints (9) implies the secondary constraints
c6"".~ 0,
8S 82~'
The associated conserved charge via first Noether theorem is QNo~,.~--
- ½ (¢*f~;,
T,~( O, p ) .~ O,
where T~ are the secondary constraints of the original action ~ written in the phase space of fields and antifields. Summing up, the first class constraints are ~z,~0, ~0,
p*i~0,
:~*~0,
T~0.
This constraint structure ensures that we have the correct number of degrees of freedom: 2 ( n - r). It is useful to express the hamiltonian H in terms of the original hamiltonian H (°~ (0, P, 2) (2):
H= H (°' -~* a ' , c6" + ~,~( - Va '~cga+f ~2~' c~.a) - ½¢'*f~r c6';'~;'e+ Hanti" 289
Volume 224, number 3
PHYSICS LETTERS B
In an analogous way one can write the conserved charge QNoether (7), associated to the BRST symmetry (6), in h a m i l t o n a i n terms ~3.
Q=T~(O,p) cg~_½~fo~ cg;,~fl_+_Qanti, where Hant~ a n d Qan~ are pieces c o n t a i n i n g terms at least quadratic on the antifields. Note that H a n d Q are defined up to primary constraints. In order to work in the same phase space of Batalin, F r a d k i n a n d Vilkovisky [ 3 ] we will introduce the canonical gauge-fixings
~*~0,
(0', p,)~ (,~%~ ) e ( ( # % ) ~ ) we have the first class constraints
To,(¢2,p)~O,
~"~0.
Note that in this space we still have gauge transform a t i o n s generated [ 11-13 ] by a c o m b i n a t i o n of the constraints n~ ~ 0 a n d T . ~ 0 given by G=n,~+
[T~ + ( f ~ 2 ; -
V~ ~)np]e ~.
The expressions of the BRST charge Q and the hamiltonian H in this phase space are
Q=T~¢"
-
" c(~ ~ p , -~ . o,f,~;,
H=HBRsT -- { ~, O},
(11)
with
/4,,~=/to-~vp~
,, ~=~,,~-.
Notice that, in spite of having introduced a term of the form { ~, Q}, the original gauge invariance (3) has not been broken. Indeed, imposing the gauge fixing constraints ~ ~ 0, we recover the original gauge invariant theory. The expression of Q given by ( 11 ) is the m i n i m a l BRST charge obtained in the standard h a m i l t o n i a n formalism [ 3,4 ]. S u m m i n g up, we have shown in the case of irreducible first rank theories the equivalence between the ~3 More details will he published elsewhere.
290
lagrangian formalism with fields a n d antifields and the h a m i l t o n i a n approach of Batalin, Fradkin and Vilkovisky. It should be stressed that the gauge invariance of the original action has been m a i n t a i n e d throughout our study. This work is partially supported by a NATO Collaborative Research G r a n t ( 0 7 6 3 / 8 7 ) . C.B. acknowledges a postdoctoral fellowship and J.P. and J.R. fellowships from Minesterio de Educaci6n y Ciencia.
~.~0,
so that p . i ~ 0 a n d ~ * ~ 0 become second class constraints. Using the associated Dirac bracket we can eliminate all these variables. In the reduced phase space
n~0,
29 June 1989
References [1 ] C. Becchi, A. Rouet and R. Stora, Ann. Phys. 98 (1976) 287; |.V. Tyutin, Lebedev preprint FIAN No. 39 (1975), unpublished. [2] P.A.M. Dirac, Lectures on quantum mechanics, Yeshiva University (New York, 1964). [3 ] E.S. Fradkin and G.A. Vilkovisky,Phys. Lett. B 55 (1975) 224; I.A. Batalin and G.A. Vilkovisky,Phys. Lett. B 69 (1977) 309; E.S. Fradkin and T.E. Fradkina, Phys. Lett. B 72 (1978) 343; I.A. Batalinand E.S. Fradkin, Phys. Lett. B 122 (1983) 157; Riv. Nuovo Cimento 9 (1986) 1. [4] M. Henneaux, Phys. Rep. 126 (1985) 1. [5] T. Kugo and I. Ojima, Suppl. Prog. Theor. Phys. 66 (1979) 1; T. Kugo and S. Uehara, Nucl. Phys. B 197 (1982) 378. [6] L. Baulieu, Phys. Rep. 129 (1985) 1. [ 7 ] I.A. Batalin and G.A. Vilkovisky,Phys. Lett. B 102 ( 1981 ) 27; Phys. Rev. D 28 (1983) 2567; I.A. Batalin and E.S. Fradkin, J. Math. Phys. 25 (1984) 2426. [8 ] J.M.L. Fisch and M. Henneaux, Homologicalperturbation theory and the algebraic structure of the antifieldantibracket formalism for gauge theories, Brussels preprint (1988). [9] C. Batlle and J. Gomis, Phys. Rev. D 38 (1988) 1169. [ 10] R. Casalbuoni, Nuovo Cimento A 33 (1976) 115. [ 11 ] L. Castellani, Ann. Phys. (NY) 143 (1982) 357. [ 12] J. Gomis, K. Kamimura and J.M. Pons, Prog. Theor. Phys. 71 (1984) 1051; C. Batlle, J. Gomis, X. Gr&cia and J.M. Pons, Noether's theorem and gauge transformations. Application to the bosonic string and CP~'-~ model, Barcelona preprint UBECM-PF 3/87, J. Math. Phys., to be published. [13] X. Gr~cia and J.M. Pons, Ann. Phys. (NY) 187 (1988) 355.
PHYSICS LETTERS B
29 June 1989
LAGRANGIAN AND HAMILTONIAN BRST FORMALISMS C. BATLLE 1, j. G O M I S 2, j. PARIS and J. R O C A
Departament d'Estructura i Constituents de la Matbria, Universitatde Barcelona, Diagonal647, E-08028Barcelona, Spain Received 14 March 1989
We establish, at the classical level, the equivalence between the standard BRST hamiltonian formalism and the lagrangian BRST approach with fields and antifields for irreducible first rank theories.
Nowadays the BRST symmetry [ 1 ] plays a crucial role in the quantization of gauge theories. At the classical level it constitutes the modern language to study constrained systems [2]. The BRST symmetry can be studied in the hamiltonian [3,4] and lagrangian [ 5 - 8 ] formalisms for irreducible and reducible systems. The aim o f this paper is to study, at the classical level, the relation between both formalisms using the approach with fields and antifields of Batalin and Vilkovisky [ 7 ] as our starting point. We will restrict ourselves to irreducible first rank theories with first class constraints described by an action ~o[¢'2"]. The associated canonical action ~ , is given by
The number o f phase space physical degrees of freedom is 2 ( n - r). The Poisson bracket algebra among T~ and Ho gives
{To,, T/j}=f~/jTy,
{Ho, T,~}= V. PT~,
w h e r e f ~ p and V, p are constant. The canonical action (1) has the gauge transformations 8 ¢ ' = { ¢ i, T.}e",
8P,={pi, To~}e",
82" = ~ " - V/j "eP + f ~,/2~'e/~. The original action ~ [ ~ , 2], with lagrangian
Lo(~, 6, 2), is invariant under 8q~'=FL~{~', T , } e " - a ' , ( ~ ,
~ : = f dt [~'p~-H~°)(O,p, 2)], i=l,...,n,
c e = l ..... r,
H ~°) (0, P, 2) = Ho (0, p) + 2 " T . (¢~,p),
82" = 6" - Vp "e~ + f ~y2 "/ea, (i) (2)
where ~', 2" and p,, n , are the coordinates and their corresponding momenta, respectively ~, and Ho is the "canonical" hamiltonian. The hamiltonian constraints are
7~..~0,
(3)
where FL~ is the pull-back of the Legendre transformation of Lo. In the lagrangian formalism of Batalin and Vilkovisky [ 7] the fields appearing in the classical action 5~oare generally denoted by ~ : ~ =~[~"],
a=l,...,m,
and they are included in a larger set o f " q u a n t u m " fields q~:
T.(O,p)~O.
{Oa} c {~A}, Present address: Department of Physics, Jadwin Hall, University of Princeton, Princeton, NJ 08540, USA. 2 Bitnet QUIM@EBUBECM1. ~ For simplicity we will consider e(Oi)=e(2") =0. The extension to odd variables is trivial. 288
6, 2)e",
A = I ..... M.
To each field ~ A we associate an antifield qb] with opposite statistics: ~(qbA) = ~A, E(qb3) =~A + 1.
0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division)
Volume 224, number 3
PHYSICS LETTERSB
In the space of fields and antifields one can define an odd symplectic structure called antibracket ~2: O~F O~G (F, G) =- Oq~A Oq~
0r F O~G 0~3 Oq)A"
hamiltonian formalism we need to perform the Dirac hamiltonian analysis [2 ] of the action S (5). With this purpose we consider the momenta associated to the fields and antifields,
OL
The "quantum" action S [ ~ , ~ * ] obeys the master equation (S,S)=O.
(4)
29 June 1989
OLo
OL ~ " - 0,~" =0,
OaJ.
p . i _ O~L
O~L lr*~= ~ =0,
The so-called proper solution contains M independent gauge transformations. In this space there is a natural definition of a generalized BRST symmetry [ 9 ]:
From (8) we get the following primary constraints:
a ~ A = (-- 1 ),~,(q)A, S),
~'~0,
a~.~ = ( -- 1 ),A+l ( ~ ,
.~,~÷A* ~ 0,
S).
~=
0~L o~ ~ --~;'
p*i~O,
~,~_
0L - o~---~ =o.
(8)
,~*a~O,
~*~0.
(9) (10)
Since S[@, ~ * ] obeys the master equation (4) the BRST transformation is nilpotent; furthermore, it is a symmetry of S. The proper solution of the master equation, associated to the original action 5eo[0, 2] mentioned above, reads
With the introduction of a graded symplectic structure [ 10] on the phase space of fields and antifields we find that the constraints (10) are second class. Using their associated Dirac bracket [2] we eliminate the canonically conjugate pairs (2*, ~*~). The canonical hamiltonian H is given by
S= f dt[Lo+OTa'~ ~
H = f ' p , - Lo(¢, f, 2) - ¢* a'~ (~, f, 2) ~ + ~ , ( - V~ ~ ~ P + f ~y2y cg~)
+½ ~,, fo~ c g ~ ] .
(5)
This action is invariant under the following BRST transformation: {3Q,=aia ~a, 82~= ~ ~a=
8S 60.__ aQ i'
V~ ~ ' ~ + f ~ r 2 r ~z,, 82" =
t a _~f~
;.
rg~, 8 ~ * - -
~S ~Scg.. .
(6)
+ ~1 2•~ f a~
0 02 ~ (Lo + ¢ * a ~ ~ ) ~'~ (gT~;~
.
(7)
In order to make contact with the standard BRST "~ Summation over discrete and continuous indices is assumed.
~';' ~,~,
where f i are the hamiltonian expression of the velocities 0i obtained through a partial inversion of the Legendre transformation. The stability of the primary constraints (9) implies the secondary constraints
c6"".~ 0,
8S 82~'
The associated conserved charge via first Noether theorem is QNo~,.~--
- ½ (¢*f~;,
T,~( O, p ) .~ O,
where T~ are the secondary constraints of the original action ~ written in the phase space of fields and antifields. Summing up, the first class constraints are ~z,~0, ~0,
p*i~0,
:~*~0,
T~0.
This constraint structure ensures that we have the correct number of degrees of freedom: 2 ( n - r). It is useful to express the hamiltonian H in terms of the original hamiltonian H (°~ (0, P, 2) (2):
H= H (°' -~* a ' , c6" + ~,~( - Va '~cga+f ~2~' c~.a) - ½¢'*f~r c6';'~;'e+ Hanti" 289
Volume 224, number 3
PHYSICS LETTERS B
In an analogous way one can write the conserved charge QNoether (7), associated to the BRST symmetry (6), in h a m i l t o n a i n terms ~3.
Q=T~(O,p) cg~_½~fo~ cg;,~fl_+_Qanti, where Hant~ a n d Qan~ are pieces c o n t a i n i n g terms at least quadratic on the antifields. Note that H a n d Q are defined up to primary constraints. In order to work in the same phase space of Batalin, F r a d k i n a n d Vilkovisky [ 3 ] we will introduce the canonical gauge-fixings
~*~0,
(0', p,)~ (,~%~ ) e ( ( # % ) ~ ) we have the first class constraints
To,(¢2,p)~O,
~"~0.
Note that in this space we still have gauge transform a t i o n s generated [ 11-13 ] by a c o m b i n a t i o n of the constraints n~ ~ 0 a n d T . ~ 0 given by G=n,~+
[T~ + ( f ~ 2 ; -
V~ ~)np]e ~.
The expressions of the BRST charge Q and the hamiltonian H in this phase space are
Q=T~¢"
-
" c(~ ~ p , -~ . o,f,~;,
H=HBRsT -- { ~, O},
(11)
with
/4,,~=/to-~vp~
,, ~=~,,~-.
Notice that, in spite of having introduced a term of the form { ~, Q}, the original gauge invariance (3) has not been broken. Indeed, imposing the gauge fixing constraints ~ ~ 0, we recover the original gauge invariant theory. The expression of Q given by ( 11 ) is the m i n i m a l BRST charge obtained in the standard h a m i l t o n i a n formalism [ 3,4 ]. S u m m i n g up, we have shown in the case of irreducible first rank theories the equivalence between the ~3 More details will he published elsewhere.
290
lagrangian formalism with fields a n d antifields and the h a m i l t o n i a n approach of Batalin, Fradkin and Vilkovisky. It should be stressed that the gauge invariance of the original action has been m a i n t a i n e d throughout our study. This work is partially supported by a NATO Collaborative Research G r a n t ( 0 7 6 3 / 8 7 ) . C.B. acknowledges a postdoctoral fellowship and J.P. and J.R. fellowships from Minesterio de Educaci6n y Ciencia.
~.~0,
so that p . i ~ 0 a n d ~ * ~ 0 become second class constraints. Using the associated Dirac bracket we can eliminate all these variables. In the reduced phase space
n~0,
29 June 1989
References [1 ] C. Becchi, A. Rouet and R. Stora, Ann. Phys. 98 (1976) 287; |.V. Tyutin, Lebedev preprint FIAN No. 39 (1975), unpublished. [2] P.A.M. Dirac, Lectures on quantum mechanics, Yeshiva University (New York, 1964). [3 ] E.S. Fradkin and G.A. Vilkovisky,Phys. Lett. B 55 (1975) 224; I.A. Batalin and G.A. Vilkovisky,Phys. Lett. B 69 (1977) 309; E.S. Fradkin and T.E. Fradkina, Phys. Lett. B 72 (1978) 343; I.A. Batalinand E.S. Fradkin, Phys. Lett. B 122 (1983) 157; Riv. Nuovo Cimento 9 (1986) 1. [4] M. Henneaux, Phys. Rep. 126 (1985) 1. [5] T. Kugo and I. Ojima, Suppl. Prog. Theor. Phys. 66 (1979) 1; T. Kugo and S. Uehara, Nucl. Phys. B 197 (1982) 378. [6] L. Baulieu, Phys. Rep. 129 (1985) 1. [ 7 ] I.A. Batalin and G.A. Vilkovisky,Phys. Lett. B 102 ( 1981 ) 27; Phys. Rev. D 28 (1983) 2567; I.A. Batalin and E.S. Fradkin, J. Math. Phys. 25 (1984) 2426. [8 ] J.M.L. Fisch and M. Henneaux, Homologicalperturbation theory and the algebraic structure of the antifieldantibracket formalism for gauge theories, Brussels preprint (1988). [9] C. Batlle and J. Gomis, Phys. Rev. D 38 (1988) 1169. [ 10] R. Casalbuoni, Nuovo Cimento A 33 (1976) 115. [ 11 ] L. Castellani, Ann. Phys. (NY) 143 (1982) 357. [ 12] J. Gomis, K. Kamimura and J.M. Pons, Prog. Theor. Phys. 71 (1984) 1051; C. Batlle, J. Gomis, X. Gr&cia and J.M. Pons, Noether's theorem and gauge transformations. Application to the bosonic string and CP~'-~ model, Barcelona preprint UBECM-PF 3/87, J. Math. Phys., to be published. [13] X. Gr~cia and J.M. Pons, Ann. Phys. (NY) 187 (1988) 355.