Path-integral equivalence between the extended and non-extended hamiltonian formalisms

Volume 245, number 2

PHYSICS LETTERS B

9 August 1990

Path-integral equivalence between the extended and non-extended hamiltonian formalisms Alain Dresse 1, Philippe Gregoire i and Marc Henneaux 2,3,4 Universit~ Libre de Bruxelles, Facult~ des Sciences, Campus Plaine, CP 231, B-1050 Brussels, Belgium Received 26 March 1990

A path-integral proof of the quantum equivalence between the extended hamiltonian formalism and the non-extended hamilionian formalism is given. This is done by (i) writing down the explicit relationship between the solution of the master equation for the non-extended formalism and the solution of the master equation for the extended formalism; (ii) exhibiting gauge fixing fermions for which the path-integral in the extended formalism reduces to the path-integral in the non-extended one. The equivalence of the extended and non-extended formalisms completes the explicit proof of the equivalence between the hamiltonian and lagrangian path-integrals. The discussion is formal throughout in that no attempt is made to define precisely the path integral.

Dirac's constrained h a m i l t o n i a n formulation has become a s t a n d a r d tool for the study o f gauge theories [ 1,2 ]. Starting from the lagrangian, the constraints naturally split into primary, secondary, ... constraints, and it was shown by D i r a c that the lagrangian equations are strictly equivalent to the equations o f m o t i o n derived from the " t o t a l " - or "non-extended" - hamiltonian action. Recall that the total action is the one containing only the p r i m a r y constraints i m p l e m e n t e d by means o f the Lagrange multipliers method. F o r instance, in the case o f electromagnetism, the constraints are n°~0

(primary)

(1)

and

nkk~o

(secondary).

(2)

The non-extended action reads

i Chercheur sous contrat IRSIA, Belgium. 2 Maitre de Recherches au Fonds National de la Recherche Scientifique, Belgium. 3 Also at Centro de Estudios Cientificos de Santiago, Casilla 16443, Santiago 9, Chili. 4 Bitnet address: R09706 @BBRBFU01

192

S T [ A u , ~u, u I ] = f dt d3x{ztUA u - ½[ ( n ) z + ( V X A ) z ] + Ao~k,k--UlnO} .

(3)

It is easy to check that the equations o f m o t i o n derived from ( 3 ) just become the Maxwell equations upon elimination o f the "auxiliary fields" n u a n d u 1 by means o f their equations o f motion. So, from the lagrangian point o f view, the p r i m a r y constraints are privileged. It is the total action that arises naturally. It was however realized by Dirac that: the separation o f the constraints into primary, secondary, ... constraints is not natural from the h a m i l t o n i a n point o f view, a n d that the full gauge s y m m e t r y o f the theory is revealed only if one treats all the first class constraints on the same footing [ 1 ]. In this context, one should replace the total action by the " e x t e n d e d act i o n " in which all the first class constraints, and not just the p r i m a r y ones, are explicitly enforced with their own Lagrange multiplier. In electromagnetism, this leads to

SEo[A~, ~ , u ~, u ~] = j dt d3x{nU,4 u - ½[ (zt)2+ ( V × A ) 2 ] + Ao 7~k,k -- U 17~0 - - U 27~ k,k } •

(4)

0370-2693/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. ( North-Holland )

Volume 245, number 2

PHYSICS LETTERSB

The equations of motion that follow from the extended action contain more arbitrary functions than those of the total action. Furthermore, the gauge transformations of both actions are not identical. For instance (4) is invariant under ~Ao=e

I,

~/./1 = ~ 1

~Ak-~.

notational purposes bosonic fields. In particular the canonical hamiltonian and the primary constraints are respectively denoted by Ho and G~l,). Consistency conditions lead to secondary, tertiary, ... constraints G Or2 (2) ) ~~(3) defined by 0/3 ' " ' " ~~(L) OtL {H0, ~ ~tk>~ r,,(k~Bj~O) k J = --,~kO) '-'Pj ,

--Oke2 ,

t~U 2 = ~2 ..1_o l

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(5)

and satisfying {G~i), G~) } -_ C~.~Ai)(~) n(k) • ,~,~,pj~k

whereas (3) is invariant under (5) only if e l = _ ~ 2 The physical justification of the extended formalism was given by Dirac by means of classical (i.e. non quantum) arguments [ 1 ]. More recently, the explicit link between the gauge invariances of S~ and S~" has been clarified in ref. [ 3 ]. This analysis relies on the observation that the total action can be obtained from the extended action by setting the Lagrange multipliers associated to the secondary, tertiary .... constraints (u 2 in ( 4 ) ) equal to zero. Because the extended formalism is more natural from a hamiltonian perspective, most works on hamiltonian quantization of gauge theories (operator approach, hamiltonian based path-integral, etc.) start from the extended formulation. In particular, the hamiltonian BRST charge treats all the first class constraints on an equal basis [ 4 ]. Now, alternative methods of quantization, less dependent on the hamiltonian formalism, exist. They directly derive the path-integral from the lagrangian form of the theory, without reference to the operator formalism. Because the lagrangian approach does not treat all the first class constraints symmetrically, it has been questioned whether the lagrangian-based path-integral is equivalent to the hamiltonian-based path-integral. Doubts have even been raised concerning the quantum validity of the extended formalism of Dirac. The purpose of this letter is to establish the equivalence of the extended and non-extended formulations of the theory. This is done in the context of the antifield formalism of Batalin and Vilkovisky [ 5 ]. At the heart of the proof is the main theorem given below (Theorem 1 ). For sake of simplicity, we will consider finite dimensional systems. The extension to infinite dimensional systems is formally straightforward. We follow throughout the conventions of ref. [ 3 ], and take for

(6)

(7)

For sake of simplicity, we suppose that each generation (j) = l, ..., L contains the same number of constraints, all first class ~t and independent (one generalizes easily the analysis to the more general case of reducible systems). We also make the following standard assumptions on the constraints: they are such that the V~-k)) defined in (6) vanish for k>j+ 1 and are invertible if k=j+ 1 and the structure constants CI~,))U) of formula (7) vanish for k > max (i, j ) . With these conditions the Dirac conjecture holds (see for instance ref. [ 3 ] ). The extended and total actions read respectively II

S~[z~,u~,] = J dtl~A(z)~"--Ho-.~,~..~"~ ~o),,, (8) tO

tl

SVo[Z~, u(,)] =- j dt[BA(z)2A--Ho -u°(1)G~l,)], (9) tO

where the z A, A = 1..... N stand for the canonical coordinates of the system. The Lagrange multipliers u~), j = 1..... L implementing the constraints, are independent variables in the variational principle. Denote by (~m_ (Z A, p~j) ) the set of all fields for the extended case, and ~" -- (z "~,u~l) ) for the total one. The extended action (8) is invariant under gauge transformations, which we denote symbolically as 6¢m =R~O') 8~) ,

(10)

and whose explicit form may be found in refs. [ 6,3 ]. The total action (9) is invariant under a linear combination of ( 10):

a~ If second class constraints appear in the initial lagrangian system, they can be eliminated using Dirac brackets [ 1,2 l193

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~m(L)oCtL Dm(k) IT~k(L) t,otl, { • , ~•' n~~n(L)oOtL t ~ L C,(L)s Jt'ta(L) C'(L) -~t~flk Z(k)tXLt,.(L). (ll) As shown in ref. [3], the coefficients F (a~/~)Lin ( 11 ) are determined by requiring that the gauge transformations (10) of the extended formalism preserve the gauge conditions uo)-~=0,

j>~2,

(12)

that reduce (8) to (9) ~2 It is of crucial importance to realize that while (10) contains as many independent gauge parameters e(~ ), e(2), ..., e(L) as there are first class constraints, the gauge transformations ( 11 ) of the total formalism involve only as many independent gauge parameters as there are primary first class constraints. These independent gauge parameters can be taken to be the gauge parameters e(z) associated to the last generation of constraints, since we have assumed that each generation has the same number of constraints ~3 The respective antibracket-antifield BRST treatment of both the extended and total actions involve different field spectra. Indeed, the initial sets of fields are different to begin with. Furthermore, the gauge symmetries of (8) and (9) contain different numbers of independent gauge parameters, leading to a different number of ghost variables. The solution of each master equation is given by

9 August 1990

(13) is given in ref. [ 7 ]. The q's are the ghosts related to the gauge transformations. Their ghost number is given by gh (r/) = 1, and they are fermionic. The 0 *, u*, q* 's are the antifields associated respectively to the 0, u, q's, with opposite Grassmann parity. The ghost number of an antifield, say ~*, is minus the ghost number of its associated field decremented by 1, i.e. gh(0*) = - [gh(q~) + 1 ] [5]. As already taken into account in ( 13 ) and (14) the BRST actions are subject to the respective boundary conditions (O)E E S =So,

(l) SEm¢*R~O)?](~)

(o) ST=ST,

(.~)r ,~L - = ~ - ,,~. , - -~,,,(L) ~ L ~/(L)-

,

(15)

(16)

These boundary conditions, in conjuction to the master equation (S, S) =0, fully determine the solution S up to irrelevant canonical transformations [ 5 ]. The proof of the path-integral equivalence of the two approaches will be done in two steps. (i) First we show how the solution S a of the master equation for the total formalism can be found from the solution S E of the master equation for the extended formalism. (ii) Then we show how to eliminate the undesirable variables (U *(i>~2), ~]O'~~2), q*U~L- 1) ) from the extended action, using the stationary phase method and a suitable fixing of the gauge. We will see that the gauge fixed extended action evaluated at the stationary point becomes exactly the gauge fixed total action.

s E [ ~ m, ~*rn, q(j), ~*(J) ] tl

= S E + j dt[¢mR~j * m~/)r/~) + O ( q n ) ]

(13)

to

6S ~S

for the extended case, and

8~ ~

s T [ ~ n, ~*n, ~](L), ~]*(L) ] :sT+

tl _t A ' [ z * am(L)~°tL~tW ' ma~if(L) o'tL + O ( r / q ) ]

(14)

tO

for the total case. The explicit form of the solution

#2 One can quickly check that the independence of the constraints implies the irreducibility of both (10) and ( 11 ). ,3 In the case when the number of constraints decreases with the generations, the independent gauge parameters of the total formalism are those associated to the last constraint in each "tower" o f primary, secondary, ... constraints.

194

Theorem 1. Let Sty, v*; w, w*] be a solution of the master equation for a gauge system with fields (v, w) and antifields (v*, w*), 6S ~S

0

(17)

+ ~w f ~ =

Suppose that the equation 5S/5w= 0 may be solved for w at the point w*=0 in such a way that w=w(v, v*), i.e. the w's may be considered as auxiliary fields. The master equation (17) is still satisfied on the surface Z= (w=w(v, v*), w * = 0 ) and reduces on this surface to 6S 6S ~---v~v* - - 0 .

(18)

Then the functional S defined on the surface Z = (w=w(v, v*), w * = 0 ) by S=S[v, v*; w=w(v, v*), w*=0 ] is a solution of the master equation in the variables v, v*:

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PHYSICS LETTERS B

~g ~g

---=0. ~v ~v*

(19)

Proof The proof is direct and relies on ( 17 ), the Leibnitz rule and the fact that ~S/~)w= 0 on E. We introduce the useful notations Z= (z, q(m, u* (*) ) for the sequel; these are the variables of the total formalism. Let us apply Theorem 1 to the extended action, (X, Z*) and ( ( u *(~2), r/u(L_,)), (u(,~ 2), q* (j ~ L- t ) ) ) acting respectively as the (v, v* ) and (w, w*). It is easy to check that the equations 8sE/Su *(i~2) = O, ~)sE/&I(j~Z._ 1) = 0 may be solved for (u *"~2), r/~/~_~)) at the point u.~z), r/. c/~ L- 1) =0. Accordingly,

9 August 1990

With this choice of V, one has u . ~ 2) = 5gt/~u* = 0 and so, the gauge conditions (12) are automatically en~,~'E ]~11.(i>12) = 8 S E / S u * ( i > ~ 2 ) and forced. Moreover, v~v,,v_ 8sE/8q 04 L-- l) = 8sE/8qtj~L- 1) as g/ involves only the variables Z of the total formalism, so that no extra dependence on ( u* (i~ 2 ), /~(J~
T

z T = I [ D / I ] exp(~S~,),

[D#]=[Dz]po,

(23)

where lto is the part of the measure which is not completely determined by the antifield formalism [ 5 ]. For the extended path-integral, one has - I [Du*t/) ] Z E= ff [Dx] exp ( i~ ~ / 1/j>~2

g~[x, x*] =sE[z, ~; u*">-'2)(Z,Z*), ~/~L-'~(X, Z*),

× (20)

U(i>~2) = 0 , ?]~j~
is a solution of the master equation in the fields ()~, Z*). Furthermore, the equations 6SE/~u *"~2) = 0 for qU,L-- ~) are the same, at first order, as the ones providing the gauge symmetries of the total formalism described in ref. [ 3 ]. It follows that the elimination of (u *">~2), r / ~ L - ~ ) ) by means of their own equa"~* R p~ m o ) ~,to) that tions leads to a redefinition of the v... become precisely the W .~.6.(L) ~.~ As the conditions nat~-OtL ,t ( L ) " u,<.2) = 0 reduce SoE to S T, one can conclude that ~E satisfies the boundary conditions (16) and is thus a solution of the master equation of the total formalism, ~E = S T. TO achieve our argument, it remains to compare the two path-integrals coming from the two formalisms. To perform those path-integrals, the first thing to do is to fix the gauge: let ¢, be a gauge fermion depending only of the fields Z*, and consider the gauge fixed actions (21)

S T ~---sT [X, )~"#~- 6 ~ / / / ~ ]

and

I-[

(24)

where the measure is formally determined and equal to the product over time of the Liouville measure in the extended phase space. If one evaluates the integral over (u* ,~>2), r/t/EL - ') ) by the stationary phase method, one finds that (24) becomes (23) since S T is the value of S~ at the extremum. This is a consequence of the analysis following Theorem 1. Furthermore, one gets an explicit expression of#o, which is just given by the higher h-order correction terms provided by the stationary phase method. This measure /to is such that (23) is (formally) independent on the choice of ~u, since (24) is also independent on the choice of ¢. We have thus proved the equality of (23) and (24), provided the measure/~o of the total formalism is appropriately chosen. The argument goes through unchanged if one adds the same non-minimal sector to both S E and S T. The variables of the non-minimal sector should simply be included among the (X, Z*)'s. In the case of the electromagnetism, the solutions of the master equation read respectively SE~.S

E u*(i>~2), S~,[~, r / ~ L_ ,~ ]

[Dr/(k)],

k<~L--I

E

+ J dt d3x[A*°q I --A*kOkrlZ+u*lrl I =S E

, u*= ~

u(i>~2) -- ~u* = 0 , u *(i>~2),

+u*E(02+q') l , r/U~
.

(22)

sT=sT

"~

(25)

.f dt d3x( --a*°~2--A*kOk~] 2 - U*I~2). (26) 195

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PHYSICS LETTERS B

Using the conventions above, one easily obtains SE=ST [A, A*, n, n*, u I, u *l, r/2, t/.2 ] ,

(27)

by setting, along the lines of Theorem 1, with r/~ and u .2 acting as the w's ~1 = __/~2, r]*l _...~0 , U*2 = tl*l - A * ° ,

u2=0.

(28)

The gauge fixing fermion implementing the Lorentz gauge is 9'[A, n °, u *l, ~/2] = _ ~ dtd3X(OkAkU.l) .

(29)

Eliminating the n k via their equations of motion, one has g r = f dt d3.1:( - IF~,F~"+n°O~A'- O'U*101d/2) .

(30) This leads to the usual path-integral in the Lorentz gauge: f [DAg] [D~/2] [Du*~]6(0.A ")

X exp(~ f dt d3x(-~Fj, vFUP-OUu*'Oj,rl2)) . (31) In c o n c l u s i o n , we h a v e explicitly p r o v e d the p a t h integral e q u i v a l e n c e o f the e x t e n d e d a n d n o n - e x t e n d e d f o r m a l i s m s . I f this result is c o m b i n e d w i t h t h e results o f ref. [ 8 ] o n the e l i m i n a t i o n o f the a u x i l i a r y fields a n d the results o f f e r s . [ 7,9 ] on the e q u i v a l e n c e b e t w e e n h a m i l t o n i a n B R S T m e t h o d s a n d antifields m e t h o d s for first o r d e r lagrangians, one concludes that the h a m i l t o n i a n a n d lagrangian path-integrals are also e q u i v a l e n t for a r b i t r a r y s e c o n d o r d e r lagrangians (fulfilling the s t a n d a r d regularity a s s u m p t i o n s o n v~k)~ a n d CI~))U) m e n t i o n e d a b o v e ~4 *~ The equivalence of the lagrangian and hamiltonian path-integrals has been checked recently on some specific examples [10].

196

9 August 1990

A more complete analysis of this last point, with examples, will be reported elsewhere [ 11 ].

References [1] P.A.M. Dirac, Lectures on quantum mechanics (Belfer Graduate School of Science, Yeshiva University, New York). 12 ] A. Hanson, T. Regge and C. Teitelboim, Constrained hamiltonian system (Accademia Nazionale dei Lincei, Rome, 1976). [ 3 ] M. Henneaux, C. Teitelboim and J. Zanelli, Nucl. Phys. B 332 (1990) 169. [4] E.S. Fradkin and G.A. Vilkovisky, Phys. Lett. B 55 (1975) 224; I.A. Batalin and G.A. Vilkovisky, Phys. Len. B 69 ( 1977 ) 309; E.S. Fradkin and T.E. Fradkina, Phys. Lett. B 72 (1978) 343; M. Henneaux, Phys. Rep. 126 (1985) 1. [ 5 ] I.A. Batalin and G.A. Vilkovisky, Phys. Lett. B 102 (1980) 27; Phys. Rev. D 28 ( 1983 ) 2527; J. Math. Phys. 26 ( 1985 ) 172; J.M.L. Fisch and M. Henneaux, Commun. Math. Phys. 128 (1990) 627; M. Henneaux, Lectures on the antifield-BRST formalism for gauge theories, in: Proc. GIFT Meeting, and in: Quantum Mechanics of fundamental systems 3 (Plenum, New York, 1991 ), to appear. [6] C. Teitelboim, The hamiltonian structure of spacetime, Ph.D. Thesis (Princeton, 1973 ); E.S. Fradkin and G.A. Vilkovisky, CERN Report TH-2332 (1977), unpublished. [7] J.M.L. Fisch and M. Henneaux, Phys. Lett. B 226 (1989) 80. [8 ] M. Henneaux, Phys. Lett. B 238 (1990) 299. [9] W. Siegel, Intern. J. Mod. Phys. A 4 (1989) 3951; C. Batlle, J. Gomis, J. Paris and J. Roca, Phys. Lett. B 224 (1989) 288. [ 10] C. BatUe, J. Gomis, X. Gracia and J.M. Pons, J. Math. Phys. 30 (1989) 1345; V. Del Duca, L. Magnea and P. van Nieuwenhuizen, Intern. J. Mod. Phys. A 3 (1988) 1081; see also M. Henneaux, Found. Phys. 17 (1987) 637. [ 11 ] A. Dresse, J.M.L. Fisch, Ph. Gr6goire and M. Henneaux, in preparation.