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Gauge fixing in extended phase space and path integral quantization of systems with second class constraints
Physics Letters B 305 (1993) 348-352 North-Holland
PHYSICS LETTERS B
Gauge fixing in extended phase space and path integral quantization of systems with second class constraints A . R e s t u c c i a a n d J. S t e p h a n y 1
Universidad Simon Bolivar, Departamento de Fisica, Apartadopostal 89000, Caracas 1080-A, Venezuela
Received 5 February 1992; revised manuscript received 16 August 1992
A new formulation of dynamical systems with second class constraints as pure gauge theories in an enlarged phase space is presented. The BRST invariant effective action for the gauge invariant version of the system is constructed. The explicit reduction on the path integral to the original phase space variables is done. Application to topologically massive electrodynamics in 3D is briefly discussed.
The study o f the formulation and properties o f classical and q u a n t u m constrained systems [ 1 ] has evolved into an important topic o f research. A set of interrelated hierarchies has grown which classifies the constraints o f the different models in first and second class, with open or with closed algebra and irreducible or n-reducible. With the appearance of models, which like the Green-Schwarz super-string [ 2 ] or the Brink-Schwarz super-particle [ 3 ], have mixed (i.e. not separable in a covariant manner) first and second class constraints this picture gained further complications. Systems with irreducible second class constraints were the first which showed themselves amenable to a systematic treatment. They may be handled in principle, by means o f the Dirac bracket construction, in a similar way as unconstrained systems [ 1 ]. However for systems with infinitely m a n y degrees of freedom as field theories or string and superstring models this strategy fails because the inversion of the Poisson brackets matrix leads to intractable non-local quantities. On the other hand, modern B R S T - B F V theory [4,5 ] has provided the tools for a complete and systematic handling of models with first class constraints. In particular for systems with (super) spacetime symmetries the gauge fixing freedom in the L Visiting researcher at Instituto Venezolano de Investigacione Cientificas, IVIC. 348
B R S T - B F V formalism allows to check covariance and unitarity explicitly. This situation suggests that a natural way to overcome the obstacles mentioned in relation with the formulation o f physical theories with second class constraints is to identify them as gauge fixed versions o f models with only first class constraints. This idea was investigated in refs. [6,7] where a class o f systems whose second class constraints allow a natural separation in gauge transformation generators and gauge fixing conditions was characterized. Another approach to obtain such an identification was presented in refs. [ 8,9 ]. In these papers, enlarging the phase space by inclusion of auxiliary variables others than usual ghosts, an algebra o f first class constraints was developed. Nevertheless in these works an explicit demonstration of the full equivalence of the gauge model with the original system was not given. In this letter we develop further on this approach. We present first a simplified construction o f the first class constraints and the gauge invariant Hamiltonian in the enlarged space. Then we show that for this gauge invariant model to be equivalent to the original one further constraints must be imposed. These constraints result to be second class again but with the advantage that they have c-number Poisson brackets. For these reasons when doing the path integral o f the system they can be used to integrate over half of the introduced variables and to end up with a
0370-2693/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
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PHYSICS LETTERS B
pure gauge theory. In order to establish the equivalence between both systems, we discuss explicitly the reduction to the physical degrees of freedom in the path integral formulation, showing the way from the BRST invariant effective action in the extended phase space, to the usual expression [ 10] in terms of the original variables. Let us consider a dynamical system, with second class and irreducible constraints, Ca(q,p)=O,
a = l .... , n ,
(1)
where q and p stand for the whole set of fields of the models. We shall assume for simplicity that they have even parity ~(0) = 0; the more general case of even and odd objects is similar. Following ref. [ 8 ] we extend the phase space to a more general manifold ~t'o with additional local conjugate variables ~, ~. We define (2a)
~)a = __ l ~a + o)ab~b '
with the Poisson brackets {~a, ~b}=o9ab.
(2b)
Here, 09 is a c-number invertible antisymmetric matrix and V is an invertible matrix. We look for a extension of Oa o n ~'/o with the following structure: (3a)
~a =Oa drO)bc ~ C V b ,
where tO~bo9~= $ c .
(3b)
At this point we leave the approach of ref. [ 8 ] and impose directly the condition on 0 to be first class constraints, {q~,, Oa} = - 2(C~a+
wfaedO,)fgOg)~e
(4)
with structure functions linear on ~. C and W are antisymmetric with respect to the lower indices and W also antisymmetric with respect to the upper indices. Both members in (4) are at most quadratic in ~. Using (3) we satisfy (4) if the field dependent objects V, C, Wand the c-number matrix o9 satisfy {(~a, Od} + 2~edOe - - o g b c Vba V dc,
(5a)
{Oa, v b } + { V b a , Od}+ZW~(b¢+2C~Vb~ =0 ,
(5b)
20 May 1993
{ v~, v~} + { v~, v~} + 2VeW,,d+ ~ ~ 2VeWad=O ~ .
(5c) From (5c), using the invertibility property of V, we may express Win terms of V. We may then obtain C f r o m (5b) in terms of Vand 0. We are thus left to solve (5a), which on-shell, gives the Poisson bracket of second class constraints as a quadratic expression on the field dependent object V. It is going to be this relation which will solve the problem of incorporating the detl/2{~, 0} field dependent measure of the functional integral to an integrand in the effective action. We shall assume in what follows the existence of a local solution of (5a) in terms of V. The invertibility of Vfollows then from det{0, 0} ~ 0. It is worth pointing that eq. (5) arises also in the expansion of the generalized algebra of Batalin and Fradkin [8] (see eqs. (2.13)-(2.26)). In fact (5a) is also the starting point of their construction. A gauge invariant extension of the Hamiltonian on ~ o (and of any other object) which is linear in • may be constructed. We have Ho = n o + ~chc,
(6)
and impose
{~o, ~a}= (U~a+'~cUL)~,,,
(7)
where h and the structure functions U have to be determined. We obtain, after expanding both members of (7) in terms of ~,
UdaOd={Ho, ()~}-- Vb hb ,
(8a)
o9b~{Ho, vb)+{hc, O,,}=Udwb~Vb+ud~Od,
(8b)
o9bAh~, Va~}+ogb/hc, Va~} =
d WdO.)bf+ ~ d Uac UafVdO.)bc.
(8c) There always exists a unique solution of (8). In fact, using the invertibility property of V, we may solve (8c) for Uad~, (8b) for Uaa and (8a) for hb. A similar construction may be used to obtain a gauge invariant extension of any other object. We have thus constructed a set of first class constraints (3), (4) and Hamiltonian (6), (7). From them we may construct the BRST generator 1"2and/~ extension on the super phase space with local coor349
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dinates q, p, ~, r/, C a, [.La'~[.la being the conjugate momenta to the ghost C a. We have {I2, £-2}=0,
(£2,~} = 0 ,
(9a,b)
with boundary conditions OC--~]C=O=(~a, /~l C=0 =/~0 .
(10a,b)
The operatorial quantization associated to (9) and (10) may also be performed following ref. [ 5 ], since we are dealing with a set of first class constraints only. We remark, however, that the construction in ref. [ 5 ] was obtained under the assumption of local expressions for ff and/-7. This may not be the case for the solution obtained from (5) and (8). The same problem is implicitly present in the construction of refs.
[8,9]. We now construct the BRST invariant effective action in the extended phase space and show that by a proper gauge fixing procedure it reduces to the known [ 10 ] effective action in the original phase space with local coordinates (q, p). To this end constraints (3), and their associated gauge fixing conditions are nevertheless insufficient and new restrictions ~5= 0 (defined below) must be introduced. The BRST invariant effective action is S~ff=(p~l+ItC+rl~-ITI-$(2lt)+~(C.Z)) .
(11)
20 May 1993
that now since o9ab is a c-number the contribution of the constraints (13a) to the integration measure is a numerical factor. In fact as we show below these constraints may be used to remove half of the additional degrees of freedom to end with a pure gauge theory. It is worth pointing out that these additional second class constraints are also unavoidable in the earlier approaches [8,9 ]. The BRST transformations of the fields, ghosts, antighosts and Lagrange multipliers are as given in refs. [ 5,11 ]. The BRST transformation law for ~ and q corresponds to canonical variables. From (13b) one shows that ~5 are BRST invariant. Problem ( 11 ) is then well posed. Z are the gauge fixing conditions. The functional integral is given by I = f ~ z detW2o~"~(~) exp(-Seff) , where ~ z is the Liouville measure ~ z = n ~ q ~ p ~ C @lt ~
~rI~C' ~ B ~ 2 ~ 0 ,
(15) and 0= $2 [ 5,11 ]. One can show [ 5,11 ] that 81/
~x=O. We first show that in spite of the factor det I/2~o in the measure (14) can be regarded as the effective action of a pure gauge theory. Let us write without lose of generality, ~oab= VaPcaYb c a with • d e t p = 1. Changing variables from ~ and q to • and • we have
Here 2a are the Lagrange multipliers associated to the extended constraints
D~ ~t/detl/2og"'~(~)
~-{/t, t2},
Changing from • to a new variable defined by
(lZa)
(0, ~2}=0,
01~=o=q~.
(12b)
An analysis of the dynamical content of the gauge theory defined by the effective action ( 1 1 ) shows the presence of undesidered degrees of freedom. To cut them, further constraints must be imposed on the system. An adequate set of constraints is (J~a~ l ~a.lt. ojab?]b = 0 .
(13a)
(16a)
(16b)
I= ~ ~exp(-Seff),
(17a)
with the measure ~q ~p ~C~/t ~ C
~ B ~2 ~ 0 , (17b)
{q~a, ~ b } = 0 ,
(13b,c)
and so they are again second class. What is gained is 350
det- 1 / 2 o j ~ ( ~ ) .
the Jacobian of the transformation is equal to det I/2to and cancels the non-desired factor in (16a). We end with a pure gauge theory whose path integral is given by
~=n
They satisfy {~a, ~b}=_09ab,
= ~qO ~
~a=~,[~l~ b ,
which satisfy
(14)
and the effective action
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(17C)
We have thus been able to incorporate to the effective action the potentially field dependent factor det{O, O} 1/2 of the measure. Moreover by constructing a first class extension of the original theory we may proceed directly to the operational quantization following ref. [ 5 ]. Let us now show that (14) reduces in fact to the Senjanovic-Fradkin functional integral [ 10 ] in the reduced phase space with local coordinates (q, p). We start with ( 14 ) instead of (17). As gauge fixing conditions we take X= tp. It is straightforward to check that these are admissible gauge conditions. The explicit expressions of effective action ( 11 a) in this gauge is
Let us now illustrate the method discussing the selfdual massive vectorial field in 3D [ 12 ]. This model is defined by the action
S= ½m( BuBU-~UVOBu OvBp ) ,
(18)
In (14) integration on B yields
Ho=(½mZBiBi+½(%OiBj)(ekzOkBt))
(22)
(we use q= ( + - - ), ¢m2= 1 = -%12, %=%0) with the second class constraints, that in order to retain explicit rotational invariance, one can take in the form ,
02(Y) = - % O~Tt~(y)+ ½m OkBk(y) •
(23a) (23b)
Here the n~ are the canonically conjugate momenta associates to Bj. The non-vanishing Poisson bracket reads
6(q~) J(~)detl/2to " on the measure. We have
{0, (x), ¢=(y)} = - m 0] 0 J J Z ( x - y ) .
&(q~)j(qS)_ d(~)J(r/) det co
(24)
An explicit local solution of eqs. ( 5a)- (5c) for this model is given by
We integrate on ~, r/, O,/z and obtain I = f rt ~q ~ p ~ c ~ ~2 det-1/2og
X exp ( - ( p ~ - H + 2~ + CCV) ) .
(21)
and is known [ 13 ] to be equivalent to the gauge invariant topologically massive theory [ 14 ]. Moreover in ref. [ 7 ] it was shown that these two models are in fact canonically equivalent, the self-dual being a gauge fixed version of the other. After eliminating in a local fashion the Bo field, which is a quadratic Lagrange multiplier, the Hamiltonian of the model is written in the form
(91( x ) = Oini(x ) + ½m% OiBj( x )
+Bq'+CaCbV~ >.
20 May 1993
( 19 )
Finally integration on C and C yields a factor det V in the measure. From (5a) we get, on the manifold ~ = 0 over which we are restricted in the functional integral by the integration on 2, det-1/2o9" det V=detl/2{~, ~} . We finally obtain
a al 2 Vb(x, y) =ab,~ (Xl - y ~ ) ,
(25a)
C~,~(x, y, z ) = 0 = W ~ ( x , y, z,s) ,
(25b)
o912(x, y) = - m 0] 0JJ z ( x - y ) .
(25c)
Using (25 ) and the introduced auxiliary variables and ~/b (a, b= 1, 2) the two pairs of self-conjugate field q)~ and ~a and generalized constraints ffa may be written in the form given by eqs. (2), (13a) and (3). In accordance with the general construction presented above the original system is described in the extended phase space by the Hamiltonian Ho constrained by ffa = 0 and ~a = 0./to is given by ~a
Ho = no + ( ~ C ( x ) h c ( x ) ) ,
I = f ~q ~ p ~2 detX/2{q~, 0}
(26)
and we solve (8) also in a local fashion to have Xexp( - ( p i I - H + 2 O ) ) ,
(20)
which is the known [ 10 ] functional integral on the original phase space.
Ug(x, y) = 0 = U~(x, y, z ) ,
(27a)
hc(x) = {Ho, ~c(X) }.
(27b) 351
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All c o m p u t a t i o n s m a y be p e r f o r m e d n o w a n d we check explicitly the passage from (11 ) to ( 2 0 ) . I n this case since {~a, ~b} = 0 o n e c a n r e i n t e r p r e t [7] o n e o f the ~ a as a first class c o n s t r a i n t a n d the other as a gauge fixing c o n d i t i o n a n d we get a true gauge theory i n e x t e n d e d space. D o i n g so we c a n recover the c a n o n i c a l e q u i v a l e n c e b e t w e e n self-dual a n d topologically m a s s i v e v e c t o r m o d e l s s h o w n in ref. [ 7 ]. T h e m e t h o d p r e s e n t e d in this p a p e r is quite general a n d applies to a wide class o f systems o f interest in q u a n t u m m e c h a n i c s a n d field theory. It m a y be generalized to the case o f r e d u c i b l e c o n s t r a i n t s which i n c l u d e a m o n g others the B r i n k - S c h w a r z superparticle [ 3] a n d the G r e e n - S c h w a r z superstring [ 2]. I n ref. [ 15 ] we apply the a d e q u a t e e x t e n s i o n o f the m e t h o d to the B r i n k - S c h w a r z superparticle. There we p e r f o r m the c a n o n i c a l q u a n t i z a t i o n o f this system in a c o v a r i a n t way i n c l u d i n g the c o n s t r u c t i o n o f the correct B R S T operator. T h a n k s are given to J. S t e p h a n y Zils for a careful r e a d i n g o f the m a n u s c r i p t .
References [ 1 ] P.A.M. Dirac, Lectures in quantum mechanics, Belfer Graduate School of Science (Yeshiva University, New York, 1964).
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20 May 1993
[2 ] M.B. Green and J. Schwarz, Phys. Lett. B 109 ( 1983 ) 399. [ 3 ] L. Brink and J. Schwarz, Phys. Lett. B 100 ( 1981 ) 287; E. Bergshoeffand R. Kallosh, Phys. Lett. B 240 (1990) 105; W. Siegel, Phys. Lett. B 203 (1988) 79; R. Kallosh, Phys. Lett. B 251 (1990) 134. [4] C. Becchi, A. Rouet and R. Stora, Ann. Phys. 98 (1976) 287. [ 5 ] E.S. Fradkin and G.A. Vilkovisky, Phys. Lett. B 55 ( 1975 ) 224; CERN report TH-2332 ( 1977); I. Batalin and E. Fradkin, Phys. Lett. B 122 (1983) 157; B 128 ( 1983 ) 307; Ann. Inst. Henri Poincar6 49 (1988) 215. [ 6 ] K. Harada and H. Mukaida, Z. Phys. C 48 (1990) 151. [ 7 ] R. Gianvittorio, A. Restuccia and J. Stephany, Mod. Phys. Lett. A 6 ( 1991 ) 2121. [8] I.A. Batalin and E.S. Fradkin, Phys. Len. B 180 (1983) 157; Nucl. Phys. B 279 (1987) 514. [9] I.A. Batalin, E.S. Fradkin and T.E. Fradkina, Nucl. Phys. B 314 (1989) 158; I.A. Batalin and I.V. Tyutin, Intern. J. Mod. Phys. A 6 (1991) 3255. [ 10] G. Senjanovic, Ann. Phys. 100 (1976) 227; E.S. Fradkin, Acta Universitatis Wratislaviensis N207, Proc. Xth Winter School of Theoretical Physics (Karpacz, 1973), p. 93. [ 11 ] M. Caicedo and A. Restuccia, USB preprint SB/FM/F 161. [ 12] P.K. Towsend, K. Pilch and P. van Nieuwenhuizen, Phys. Len. B 136 (1984) 38. [ 13 ] R. Jackiw and S. Deser, Phys. Lett. B 139 (1984) 371. [14JR. Jackiw, S. Deser and S. Templeton, Ann. Phys. 140 (1982) 372. [ 15 ] A. Restuccia and J. Stephany, USB preprint SB/F/92-189.
PHYSICS LETTERS B
Gauge fixing in extended phase space and path integral quantization of systems with second class constraints A . R e s t u c c i a a n d J. S t e p h a n y 1
Universidad Simon Bolivar, Departamento de Fisica, Apartadopostal 89000, Caracas 1080-A, Venezuela
Received 5 February 1992; revised manuscript received 16 August 1992
A new formulation of dynamical systems with second class constraints as pure gauge theories in an enlarged phase space is presented. The BRST invariant effective action for the gauge invariant version of the system is constructed. The explicit reduction on the path integral to the original phase space variables is done. Application to topologically massive electrodynamics in 3D is briefly discussed.
The study o f the formulation and properties o f classical and q u a n t u m constrained systems [ 1 ] has evolved into an important topic o f research. A set of interrelated hierarchies has grown which classifies the constraints o f the different models in first and second class, with open or with closed algebra and irreducible or n-reducible. With the appearance of models, which like the Green-Schwarz super-string [ 2 ] or the Brink-Schwarz super-particle [ 3 ], have mixed (i.e. not separable in a covariant manner) first and second class constraints this picture gained further complications. Systems with irreducible second class constraints were the first which showed themselves amenable to a systematic treatment. They may be handled in principle, by means o f the Dirac bracket construction, in a similar way as unconstrained systems [ 1 ]. However for systems with infinitely m a n y degrees of freedom as field theories or string and superstring models this strategy fails because the inversion of the Poisson brackets matrix leads to intractable non-local quantities. On the other hand, modern B R S T - B F V theory [4,5 ] has provided the tools for a complete and systematic handling of models with first class constraints. In particular for systems with (super) spacetime symmetries the gauge fixing freedom in the L Visiting researcher at Instituto Venezolano de Investigacione Cientificas, IVIC. 348
B R S T - B F V formalism allows to check covariance and unitarity explicitly. This situation suggests that a natural way to overcome the obstacles mentioned in relation with the formulation o f physical theories with second class constraints is to identify them as gauge fixed versions o f models with only first class constraints. This idea was investigated in refs. [6,7] where a class o f systems whose second class constraints allow a natural separation in gauge transformation generators and gauge fixing conditions was characterized. Another approach to obtain such an identification was presented in refs. [ 8,9 ]. In these papers, enlarging the phase space by inclusion of auxiliary variables others than usual ghosts, an algebra o f first class constraints was developed. Nevertheless in these works an explicit demonstration of the full equivalence of the gauge model with the original system was not given. In this letter we develop further on this approach. We present first a simplified construction o f the first class constraints and the gauge invariant Hamiltonian in the enlarged space. Then we show that for this gauge invariant model to be equivalent to the original one further constraints must be imposed. These constraints result to be second class again but with the advantage that they have c-number Poisson brackets. For these reasons when doing the path integral o f the system they can be used to integrate over half of the introduced variables and to end up with a
0370-2693/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
Volume 305, number 4
PHYSICS LETTERS B
pure gauge theory. In order to establish the equivalence between both systems, we discuss explicitly the reduction to the physical degrees of freedom in the path integral formulation, showing the way from the BRST invariant effective action in the extended phase space, to the usual expression [ 10] in terms of the original variables. Let us consider a dynamical system, with second class and irreducible constraints, Ca(q,p)=O,
a = l .... , n ,
(1)
where q and p stand for the whole set of fields of the models. We shall assume for simplicity that they have even parity ~(0) = 0; the more general case of even and odd objects is similar. Following ref. [ 8 ] we extend the phase space to a more general manifold ~t'o with additional local conjugate variables ~, ~. We define (2a)
~)a = __ l ~a + o)ab~b '
with the Poisson brackets {~a, ~b}=o9ab.
(2b)
Here, 09 is a c-number invertible antisymmetric matrix and V is an invertible matrix. We look for a extension of Oa o n ~'/o with the following structure: (3a)
~a =Oa drO)bc ~ C V b ,
where tO~bo9~= $ c .
(3b)
At this point we leave the approach of ref. [ 8 ] and impose directly the condition on 0 to be first class constraints, {q~,, Oa} = - 2(C~a+
wfaedO,)fgOg)~e
(4)
with structure functions linear on ~. C and W are antisymmetric with respect to the lower indices and W also antisymmetric with respect to the upper indices. Both members in (4) are at most quadratic in ~. Using (3) we satisfy (4) if the field dependent objects V, C, Wand the c-number matrix o9 satisfy {(~a, Od} + 2~edOe - - o g b c Vba V dc,
(5a)
{Oa, v b } + { V b a , Od}+ZW~(b¢+2C~Vb~ =0 ,
(5b)
20 May 1993
{ v~, v~} + { v~, v~} + 2VeW,,d+ ~ ~ 2VeWad=O ~ .
(5c) From (5c), using the invertibility property of V, we may express Win terms of V. We may then obtain C f r o m (5b) in terms of Vand 0. We are thus left to solve (5a), which on-shell, gives the Poisson bracket of second class constraints as a quadratic expression on the field dependent object V. It is going to be this relation which will solve the problem of incorporating the detl/2{~, 0} field dependent measure of the functional integral to an integrand in the effective action. We shall assume in what follows the existence of a local solution of (5a) in terms of V. The invertibility of Vfollows then from det{0, 0} ~ 0. It is worth pointing that eq. (5) arises also in the expansion of the generalized algebra of Batalin and Fradkin [8] (see eqs. (2.13)-(2.26)). In fact (5a) is also the starting point of their construction. A gauge invariant extension of the Hamiltonian on ~ o (and of any other object) which is linear in • may be constructed. We have Ho = n o + ~chc,
(6)
and impose
{~o, ~a}= (U~a+'~cUL)~,,,
(7)
where h and the structure functions U have to be determined. We obtain, after expanding both members of (7) in terms of ~,
UdaOd={Ho, ()~}-- Vb hb ,
(8a)
o9b~{Ho, vb)+{hc, O,,}=Udwb~Vb+ud~Od,
(8b)
o9bAh~, Va~}+ogb/hc, Va~} =
d WdO.)bf+ ~ d Uac UafVdO.)bc.
(8c) There always exists a unique solution of (8). In fact, using the invertibility property of V, we may solve (8c) for Uad~, (8b) for Uaa and (8a) for hb. A similar construction may be used to obtain a gauge invariant extension of any other object. We have thus constructed a set of first class constraints (3), (4) and Hamiltonian (6), (7). From them we may construct the BRST generator 1"2and/~ extension on the super phase space with local coor349
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dinates q, p, ~, r/, C a, [.La'~[.la being the conjugate momenta to the ghost C a. We have {I2, £-2}=0,
(£2,~} = 0 ,
(9a,b)
with boundary conditions OC--~]C=O=(~a, /~l C=0 =/~0 .
(10a,b)
The operatorial quantization associated to (9) and (10) may also be performed following ref. [ 5 ], since we are dealing with a set of first class constraints only. We remark, however, that the construction in ref. [ 5 ] was obtained under the assumption of local expressions for ff and/-7. This may not be the case for the solution obtained from (5) and (8). The same problem is implicitly present in the construction of refs.
[8,9]. We now construct the BRST invariant effective action in the extended phase space and show that by a proper gauge fixing procedure it reduces to the known [ 10 ] effective action in the original phase space with local coordinates (q, p). To this end constraints (3), and their associated gauge fixing conditions are nevertheless insufficient and new restrictions ~5= 0 (defined below) must be introduced. The BRST invariant effective action is S~ff=(p~l+ItC+rl~-ITI-$(2lt)+~(C.Z)) .
(11)
20 May 1993
that now since o9ab is a c-number the contribution of the constraints (13a) to the integration measure is a numerical factor. In fact as we show below these constraints may be used to remove half of the additional degrees of freedom to end with a pure gauge theory. It is worth pointing out that these additional second class constraints are also unavoidable in the earlier approaches [8,9 ]. The BRST transformations of the fields, ghosts, antighosts and Lagrange multipliers are as given in refs. [ 5,11 ]. The BRST transformation law for ~ and q corresponds to canonical variables. From (13b) one shows that ~5 are BRST invariant. Problem ( 11 ) is then well posed. Z are the gauge fixing conditions. The functional integral is given by I = f ~ z detW2o~"~(~) exp(-Seff) , where ~ z is the Liouville measure ~ z = n ~ q ~ p ~ C @lt ~
~rI~C' ~ B ~ 2 ~ 0 ,
(15) and 0= $2 [ 5,11 ]. One can show [ 5,11 ] that 81/
~x=O. We first show that in spite of the factor det I/2~o in the measure (14) can be regarded as the effective action of a pure gauge theory. Let us write without lose of generality, ~oab= VaPcaYb c a with • d e t p = 1. Changing variables from ~ and q to • and • we have
Here 2a are the Lagrange multipliers associated to the extended constraints
D~ ~t/detl/2og"'~(~)
~-{/t, t2},
Changing from • to a new variable defined by
(lZa)
(0, ~2}=0,
01~=o=q~.
(12b)
An analysis of the dynamical content of the gauge theory defined by the effective action ( 1 1 ) shows the presence of undesidered degrees of freedom. To cut them, further constraints must be imposed on the system. An adequate set of constraints is (J~a~ l ~a.lt. ojab?]b = 0 .
(13a)
(16a)
(16b)
I= ~ ~exp(-Seff),
(17a)
with the measure ~q ~p ~C~/t ~ C
~ B ~2 ~ 0 , (17b)
{q~a, ~ b } = 0 ,
(13b,c)
and so they are again second class. What is gained is 350
det- 1 / 2 o j ~ ( ~ ) .
the Jacobian of the transformation is equal to det I/2to and cancels the non-desired factor in (16a). We end with a pure gauge theory whose path integral is given by
~=n
They satisfy {~a, ~b}=_09ab,
= ~qO ~
~a=~,[~l~ b ,
which satisfy
(14)
and the effective action
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PHYSICSLETTERSB
(17C)
We have thus been able to incorporate to the effective action the potentially field dependent factor det{O, O} 1/2 of the measure. Moreover by constructing a first class extension of the original theory we may proceed directly to the operational quantization following ref. [ 5 ]. Let us now show that (14) reduces in fact to the Senjanovic-Fradkin functional integral [ 10 ] in the reduced phase space with local coordinates (q, p). We start with ( 14 ) instead of (17). As gauge fixing conditions we take X= tp. It is straightforward to check that these are admissible gauge conditions. The explicit expressions of effective action ( 11 a) in this gauge is
Let us now illustrate the method discussing the selfdual massive vectorial field in 3D [ 12 ]. This model is defined by the action
S= ½m( BuBU-~UVOBu OvBp ) ,
(18)
In (14) integration on B yields
Ho=(½mZBiBi+½(%OiBj)(ekzOkBt))
(22)
(we use q= ( + - - ), ¢m2= 1 = -%12, %=%0) with the second class constraints, that in order to retain explicit rotational invariance, one can take in the form ,
02(Y) = - % O~Tt~(y)+ ½m OkBk(y) •
(23a) (23b)
Here the n~ are the canonically conjugate momenta associates to Bj. The non-vanishing Poisson bracket reads
6(q~) J(~)detl/2to " on the measure. We have
{0, (x), ¢=(y)} = - m 0] 0 J J Z ( x - y ) .
&(q~)j(qS)_ d(~)J(r/) det co
(24)
An explicit local solution of eqs. ( 5a)- (5c) for this model is given by
We integrate on ~, r/, O,/z and obtain I = f rt ~q ~ p ~ c ~ ~2 det-1/2og
X exp ( - ( p ~ - H + 2~ + CCV) ) .
(21)
and is known [ 13 ] to be equivalent to the gauge invariant topologically massive theory [ 14 ]. Moreover in ref. [ 7 ] it was shown that these two models are in fact canonically equivalent, the self-dual being a gauge fixed version of the other. After eliminating in a local fashion the Bo field, which is a quadratic Lagrange multiplier, the Hamiltonian of the model is written in the form
(91( x ) = Oini(x ) + ½m% OiBj( x )
+Bq'+CaCbV~ >.
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( 19 )
Finally integration on C and C yields a factor det V in the measure. From (5a) we get, on the manifold ~ = 0 over which we are restricted in the functional integral by the integration on 2, det-1/2o9" det V=detl/2{~, ~} . We finally obtain
a al 2 Vb(x, y) =ab,~ (Xl - y ~ ) ,
(25a)
C~,~(x, y, z ) = 0 = W ~ ( x , y, z,s) ,
(25b)
o912(x, y) = - m 0] 0JJ z ( x - y ) .
(25c)
Using (25 ) and the introduced auxiliary variables and ~/b (a, b= 1, 2) the two pairs of self-conjugate field q)~ and ~a and generalized constraints ffa may be written in the form given by eqs. (2), (13a) and (3). In accordance with the general construction presented above the original system is described in the extended phase space by the Hamiltonian Ho constrained by ffa = 0 and ~a = 0./to is given by ~a
Ho = no + ( ~ C ( x ) h c ( x ) ) ,
I = f ~q ~ p ~2 detX/2{q~, 0}
(26)
and we solve (8) also in a local fashion to have Xexp( - ( p i I - H + 2 O ) ) ,
(20)
which is the known [ 10 ] functional integral on the original phase space.
Ug(x, y) = 0 = U~(x, y, z ) ,
(27a)
hc(x) = {Ho, ~c(X) }.
(27b) 351
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PHYSICS LETTERS B
All c o m p u t a t i o n s m a y be p e r f o r m e d n o w a n d we check explicitly the passage from (11 ) to ( 2 0 ) . I n this case since {~a, ~b} = 0 o n e c a n r e i n t e r p r e t [7] o n e o f the ~ a as a first class c o n s t r a i n t a n d the other as a gauge fixing c o n d i t i o n a n d we get a true gauge theory i n e x t e n d e d space. D o i n g so we c a n recover the c a n o n i c a l e q u i v a l e n c e b e t w e e n self-dual a n d topologically m a s s i v e v e c t o r m o d e l s s h o w n in ref. [ 7 ]. T h e m e t h o d p r e s e n t e d in this p a p e r is quite general a n d applies to a wide class o f systems o f interest in q u a n t u m m e c h a n i c s a n d field theory. It m a y be generalized to the case o f r e d u c i b l e c o n s t r a i n t s which i n c l u d e a m o n g others the B r i n k - S c h w a r z superparticle [ 3] a n d the G r e e n - S c h w a r z superstring [ 2]. I n ref. [ 15 ] we apply the a d e q u a t e e x t e n s i o n o f the m e t h o d to the B r i n k - S c h w a r z superparticle. There we p e r f o r m the c a n o n i c a l q u a n t i z a t i o n o f this system in a c o v a r i a n t way i n c l u d i n g the c o n s t r u c t i o n o f the correct B R S T operator. T h a n k s are given to J. S t e p h a n y Zils for a careful r e a d i n g o f the m a n u s c r i p t .
References [ 1 ] P.A.M. Dirac, Lectures in quantum mechanics, Belfer Graduate School of Science (Yeshiva University, New York, 1964).
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[2 ] M.B. Green and J. Schwarz, Phys. Lett. B 109 ( 1983 ) 399. [ 3 ] L. Brink and J. Schwarz, Phys. Lett. B 100 ( 1981 ) 287; E. Bergshoeffand R. Kallosh, Phys. Lett. B 240 (1990) 105; W. Siegel, Phys. Lett. B 203 (1988) 79; R. Kallosh, Phys. Lett. B 251 (1990) 134. [4] C. Becchi, A. Rouet and R. Stora, Ann. Phys. 98 (1976) 287. [ 5 ] E.S. Fradkin and G.A. Vilkovisky, Phys. Lett. B 55 ( 1975 ) 224; CERN report TH-2332 ( 1977); I. Batalin and E. Fradkin, Phys. Lett. B 122 (1983) 157; B 128 ( 1983 ) 307; Ann. Inst. Henri Poincar6 49 (1988) 215. [ 6 ] K. Harada and H. Mukaida, Z. Phys. C 48 (1990) 151. [ 7 ] R. Gianvittorio, A. Restuccia and J. Stephany, Mod. Phys. Lett. A 6 ( 1991 ) 2121. [8] I.A. Batalin and E.S. Fradkin, Phys. Len. B 180 (1983) 157; Nucl. Phys. B 279 (1987) 514. [9] I.A. Batalin, E.S. Fradkin and T.E. Fradkina, Nucl. Phys. B 314 (1989) 158; I.A. Batalin and I.V. Tyutin, Intern. J. Mod. Phys. A 6 (1991) 3255. [ 10] G. Senjanovic, Ann. Phys. 100 (1976) 227; E.S. Fradkin, Acta Universitatis Wratislaviensis N207, Proc. Xth Winter School of Theoretical Physics (Karpacz, 1973), p. 93. [ 11 ] M. Caicedo and A. Restuccia, USB preprint SB/FM/F 161. [ 12] P.K. Towsend, K. Pilch and P. van Nieuwenhuizen, Phys. Len. B 136 (1984) 38. [ 13 ] R. Jackiw and S. Deser, Phys. Lett. B 139 (1984) 371. [14JR. Jackiw, S. Deser and S. Templeton, Ann. Phys. 140 (1982) 372. [ 15 ] A. Restuccia and J. Stephany, USB preprint SB/F/92-189.