henneaux. hamiltonian from the path integral for theories with gauge freedom

HAMILTONIAN FORM OF THE PATH INTEGRAL FOR THEORIES WITH A GAUGE FREEDOM Marc HENNEAUX Center for Theoretical Physics, The University of Texas at Austin, Austin, Texas 78712, U.S.A... 4

HAMILTONIAN FORM OF THE PATH INTEGRAL FOR THEORIES WITH A GAUGE

FREEDOM

Marc HENNEAUX

Center for Theoretical Physics, The University of Texas at Austin, Austin, Texas 78712, U.S.A

NORTH-HOLLAND-AMSTERDAM

PHYSICS REPORTS (Review Section of Physics Letters) 126, No 1 (1985) 1-66 North-Holland, Amsterdam

HAMILTONIAN FORM OF THE PATH INTEGRAL FOR

THEORIES WITH A GAUGE FREEDOM

2 Constrained Hamiltonian systems 5 8 Quantum theory - general aspects 32

3 The interior product operators 5 and 83 7 8.1 Formal considerations 32 3.1 Definitions 7 8.2 Unitarity in the physical subspace 34 3.2 Main property of 52 and 53 y 8.3 Equivalence with Dirac quantization 34 3.3 Equations involving 52 and 53 1] 8.4 Equivalence with reduced phase space quantization 37

4 Structure of constrained Hamiltonian systems H 9 Path integral - Fradkin and Vilkovisky theorem 38 4.1 Second-order structure functions II 9.1, The boundary conditions for the physical states 38 4.2 Third-order structure functions I2 9.2 Fradkin and Vilkovisky theorem 41

4.4 Rank of a set of structure functions 16 10 The Yang-Mills field 44

5 The generator 2 of the Becchi-Rouet-Stora (BRS) trans- 10.1 Canonical formulation 44

5.2 The generator 2 17 10.4, The ‘Gaussian average” representation 47 5.3 Comments 20 10.5 Gauges with an ''extra ghost` 47

6.2 Existence of BRS invariant observables 2 11.2 The BRS transformation 50

7.1, A simple representation of the constraints 24 12 Reducible theories S|

7.2 The structure functions and the integrability conditions 25 12.1 Structure functions ~ BRS transformation SI 7.3 The Hamiltonian in involution with the constraints 27 12.2, Quantum theory 37 7.4 The transformation which turns the constraints into Appendix: Explicit form of the higher order structure func-

7.5 The transformation which turns the Hamiltonian into a References 62 function identically in involution with the constraints 30 Additional notes and references 64

*“Chercheur qualifié’ au Fonds National Belge de la Recherche Scientifique

On leave from Faculté des Sciences, Université Libre de Bruxelles, C.P 231, B-10S0 Bruxelles, Belgium

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Copies of this Issue may be obtained at the price given below, AlL orders should be sent directly tọ the Publisher_Orders must be

Abstract:

The Hamiltonian form of the path integral for theories with a gauge freedom is reviewed along the lines developed by Batalin, Fradkin and Vilkovisky The formalism, which can be applied to gauge theories with an open algebra without the need for auxiliary fields, heavily relies on the canonical formulation of the Becchi-Rouet-Stora transformation This transformation appears here as a purely classical object associated with the remarkable classical structure of (first class) constrained Hamiltonian systems The occurrence of multi-ghost interactions in the effective quantum action is naturally predicted

The formalism is also extended to “reducible” gauge theories, i.e., theories whose gauge transformations are not independent, within which scope the recently studied anti-symmetric gauge fields fall Here again, the BRS transformation plays a key role

1 Introduction

Since the advent of general relativity, ‘“gauge invariance” has played a key role in building models of

physical theories First restricted to true infinite-dimensional groups of transformations, the gauge

invariance concept has evolved, with the birth of supersymmetry and supergravity, to the more elaborate notion of invariance under infinitesimal transformations whose commutators do not neces- sarily close Only closure on shell of the gauge “algebra” is required and the structure “constants” may

even be field-dependent (these and other aspects of supergravity are reviewed in [57, 58])

When computing the transition amplitude between two gauge field configurations by path integral

methods, one faces the problem that the sum over trajectories diverges because of the degeneracy of

the Lagrangian It has been shown in the sixties how this question can be solved when the gauge transformations form a true group [20, 27, 29] The method is, however, on a less firm basis when the structure constants involve the fields (the question has been analyzed recently in [3]) Furthermore, it yields an incorrect answer when the algebra of the infinitesimal transformations does not close: the S-matrix fails to be both gauge invariant and unitary (see e.g [32,59]) One then needs to turn to

different approaches

One of these approaches introduces auxiliary fields in the theory, i.e., fields with no dynamics, which

appear in the modified gauge transformation laws in such a way that these become closed [57, 58}

TIOWCVC€C D€TI€ NO ema Wd O nnd hese auxi lan C1G NOT € en known 3 ney aiwad

exist (cf problems w: with N >2 2 supergravity {et 68)

treated here.) V We vill refer to this 5 approach as the > Batalin-Fradkin-Vilkovigky (BEV) method The

main goal of this report is to describe it Because of its importance, we feel that a pedagogical

exposition, which brings together the results of [4, 8, 31-34], sheds on them a different light and gives

explicit demonstrations of the theorems, is necessary

The main features of the BFV approach can be outlined as follows

(i) It does not require the closure off-shell of the gauge algebra and hence, it does not need auxiliary

fields Of course, this does not mean that auxiliary fields are not helpful, when they exist, in other

respects (simplifying some calculations .) But, in principle, they are not necessary,

here an intrinsic meaning, i e., it is ‘Independent of the gauge conditions Besides, it appears as a purely

Iti is actually fair to believe that if the study of mechanics over Grassmann algebras had been of interest

Ot} +^ 7 L he-BR 4 nofo s ry q11 ata oi h d L) avevoba ry : aAanfan thorn C Ù Kho at @ a Ace O y e d a C

to the theory of function groups developed by Lie, and others

4 Marc Henneaux, Hamiltonian form of the path integral for theories with a gauge freedom

(ii) BFV method is applicable to Lagrangians which are not quadratic in the velocities and it is hence more general than a strict Lagrangian approach

(iv) The method also applies to general gauge breaking terms in the effective Lagrangian, even those

that include the ghosts

(v) Being based on the Hamiltonian, the approach is closer to Hilbert space techniques and to

unitarity

(vi) Finally BFV method yields a definite expression for the local measure in the functional integral,

whereas the Lagrangian approach does not in the general case [9] It is true that this local measure is irrelevant in renormalizable gauge theories, where it can be set equal to one by appropriate regularization [43] But for gauge theories that are not renormalizable and of which we have now little understanding, that

local measure may play a more significant role [46]

The paper is organized along the following lines In the next section, the Dirac Hamiltonian theory of constrained systems is briefly reviewed, since gauge theories fall within its scope That survey is intended mainly to fix the notation

The analysis of the classical structure of constrained systems is then undertaken (sections 3 through 7) The structure functions are defined It is shown that they obey a series of identities from which the existence of a nilpotent transformation of the supersymmetry type in an appropriately extended phase space involving new variables, the ‘‘ghosts’’, is easily inferred This transformation reduces to the Becchi- Rouet—Stora transformation in the Yang-Mills case and is therefore called the BRS transformation

The remarkable classical structure of constrained systems, encapsuled in the BRS transformation, has gone unnoticed so far, because it is trivial for the usual gauge systems which have a closed algebra

However, it is non-trivial for theories with an open algebra, where it plays a critical role The canonical generator of the BRS transformation contains then multi-ghost terms of order higher than three

The properties of classical observables, i.e., gauge invariant functions, are also investigated We prove the existence, for any observable, of appropriate BRS invariant extensions involving the ghost variables This guarantees in particular the existence of a BRS invariant Hamiltonian

Let us stress again that the analysis of the first sections is entirely classical As we mentioned, BRS

transformations could have been invented long ago, since they simply reflect the remarkable classical

properties of constrained theories This point is particularly emphasized in section 7, where the ambiguity in the structure functions is also discussed and shown to lead to a canonical transformation in

The general properties of the quantum theory (physical subspace, observables, unitarity, .) are

transformations,” in the sense that a state which is obtained from another by application of the BRS

The path integral is then considered (section 9) Fradkin and Vilkovisky theorem, which establishes the gauge invariance of the S-matrix, is demonstrated, following a method due to Batalin and Vilkovisky The BRS invariant effective action is shown to include in general multi-ghost interactions of order higher than two It 1s here that there 1s a discrepancy with the Faddeev—Popov method, which only yields ghost—ghost interactions of order two

The power of the BFV approach is then illustrated in the Yang-Mills case (section 10), for which

standard results are derived in two lines Gauges involving extra ghosts [60, 63, 37] are also discussed

We then turn to the quantization of the relativistic membrane This model is interesting here because

it does not fall within the scope of the Faddeev-Popov method Yet, it is simple enough that the structure functions and the generator of the BRS transformation can be explicitly computed The

effective action is written and contains multi-ghost interactions of order 2n, where n is the number of spatial dimensions of the membrane [38]

Finally, we discuss the quantization of “reducible” gauge theories, i.e., theories for which not all

gauge transformations are independent These include the theories based on completely antisymmetric

tensor fields which play an important role in supergravity

The material and the presentation of this report are based in part on the author’s “‘thése d’agre-

gation” [39] Other reviews of the BFV approach, limited to theories which only need ghost-ghost

interactions of order two, can be found in refs [55, 56, 69)

2 Constrained Hamiltonian systems

Because of the gauge invariance of the action, the Legendre transformation between the generalized velocities and the canonically conjugate momenta Is singular in the case of gauge theories Hence, there

are constraints in the canonical formulation How to handle this problem without fixing the gauge has been shown by Dirac [22, 23] Good reviews exist on the subject [24, 36, 53, 69] and we will assume here that the reader is familiar with the Dirac formalism

The phase space variables, among which some may belong to a Grassman algebra [12, 13, 15], will be denoted by z,4 (A =1, ,n) The constraint functions are ¢, (za) (a = 1, , m) and their associated

Lagrange multipliers are A* We assume that the constraints are real They can be either even (€, = 0)

or odd (e, = 1) In that latter case, A* is imaginary and odd, so that A“, is both real and even (e, is the Grassmann parity of ¢,, defined by FG = (-)***°GF for any functions F, G of definite parity.)

We will also assume that the constraints are all first class, i.e., that the second class constraints (if

any) have been eliminated by means of Dirac procedure For simplicity, we take furthermore the

Hamiltonian variables z, to be real and to have canonical Dirac brackets ([z,, Ze] = Cap, det Can# 0,

Cap constants, C 4p =— Cpa), aS it is always permitted, at least locally, by Darboux theorem (this

fhe discussion below would not be more complicated _in its spirit had one not go id of the second

class constraints These would enter the functional integral via delta functions, so that one is effectively

Asta PO AfA c in nar++o/} hh he AON 2 7 ¬ Vr r ann Tatarmnin ont wrapis anrnanq r+ 1 C=r-ri = ECS a

q wy LÍ rÌ J uo Uy UU Ul q L.) a > a LG đ QÌ a WUT apy d

, ,Ố

the measure to guarantee that one is using the Liouville measure associated with the Dirac bracket It is

only for the sake of notational convenience, and because we are interested in constraints resulting fron

a true gauge freedom, that we assume here that the second class constraints have been eliminated

where H,(z,) is the classical Hamiltonian

The Poisson bracket [F, G] of two functions F, G reads, in terms of right and left derivatives,

6 Marc Henneaux, Hamiltonian form of the path integral for theories with a gauge freedom (with dF = (d’F/dz,4) dz, = dz,(0'F/0z,)) It fulfills the following algebraic useful properties, prop

[F G,Go] = [F, G,JGo+ (-) "GF, G3] (2.4b)

([Fi, Fo), Fal + (er [F, BỊ], Fil + (Oy 80s, Fi), Fa] = 0 Œ-4c)

(‘Jacobi identity’’), as well as

Elementary computations show that the structure functions obey

(V*%,)* = V*a(-) 1%, e{ V*ạ) = &a + Ca (2.5c) (There is no summation over an index which is only repeated in a Grassmann parity symbol.)

Although the brackets of ¢, and dg do close (this is simply the first class property which reflects the

gauge invariance of the theory, the constraints are preserved under a gauge transformation and hence,

; 0 infinitesimat transformations generated by @,

ðF = [F, e*®$„] = () Te*[F ba] (2.6)

is open when the structure functions C“,, involve the fields Indeed, one gets in that case

= (-)°"" e°n*C cal F dy | + (=) re en | F, C "pl, - (2 7)

Hence, it is only when the constraints hold (¢, = 0) that the transformation (2.6) close and generate an

m-dimensional submanifold: in the open case, the equations [F,¢,]= 0 which express that F is

“strongly” invariant under gauge transformations, imply further conditions on F There is accordingly

no contradiction between (2.1) and the open algebra assumption First class constrained Hamiltonian systems embrace all gauge models, with or without closed algebra

*Note, however, that by redefinition of the constraints, one can change y g their algebra almost at will (see section 7)

The canonical action is given by

plus, possibly, appropriate boundary terms at ¢, and í which we will never write explicitly [35] Ít isinvariant

under the transformations (2.6) provided one transforms the Lagrange multipliers as

These momenta are classically constrained to vanish, in order not to change the dynamical content of the theory The constraints 7, = 0 generate the gauge transformations A* > A“ + u* of the multipliers, thereby expressing their arbitrariness The introduction of these “fake” degrees of freedom makes

easier the comparison with the Lagrangian form of the path integral when the multipliers A* appear as original Lagrangian variables

since ¢, is independent of A* Most of the classical work which follows is based on the eqs (2.11a) and

Dì and actua ndenendent o he 3A nat m an even intece

In order to study the structure of constrained Hamiltonian systems, one needs to develop new tools

One such tool is given by the operators 6, and 63 acting on functions F “~~ of the canonical variables

which are completely antisymmetric in the indices a; (these functions are loose alled_“‘antisymmetri

8 Marc Henneaux, Hamiltonian form of the path integral for theories with a gauge freedom

E2P0)'4P@) = (—)?Œ al, si 84) đị dạ (3.1.1)

for any permutation P of the indices 1, , g Here, (—)?"'*" >” is the elementary sign by which the

products Y,, - Pa, and Pap°** Pap, differ (@ € Z2) The variables Y, are auxiliary variables whose

Grassmann parity is just opposite to the one of the constraint functions G,, e(P,) = e(G,)+ l=e,t+ 1

[If an anticommuting Y,, appears twice (or more) in the product Y,, - P,,, one sets F “'"* = 0.] From the point of view of the variables Y,, one could say that F “'“ is actually symmetric, but we prefer to stick to the terminology “antisymmetric,” for this is what antisymmetry as defined above amounts to

when all constraints are even

If the permutation is just a transposition of ¡7 and 7 ø reads explicitly

Đự — (£; + Me; + 1)+ (£; — £j)Hụ (mod.2) 3 (3.1.2)

with ¢; = €,, and n, is the number of even indices between a; and q; For instance one finds, when F is

a 2-index tensor, that

a1 = — Fa!

(3.1.3a)

if a, and a, both correspond to even constraints and that

pas pom (3.1.3b)

if one of the a; is odd It is this definition of antisymmetry that will prove useful in the sequel

(the notation 62 was introduced in [56] in the commuting case) 6, is very similar to the interior

product operator of a q-form by a vector Similarly, one defines 63 by

One finds that

One goes from the first line of (3.1.9) to the second by using the antisymmetry of F;, from the second to

the third by expanding the exponent of minus one and by relabelling the indices, and from the third to

the fourth by using Gz,_,Ga, = Ga,Ga,-:(—)°*"*""! (We have used the convention that 62 (or 63) applied

to a function, i.e., to a tensor of order zero, yields its restriction on the constraint surface Functions

defined only on that surface can in turn be thought of as being mapped on zero.)

If all the constraints are even, 52 = 63 If all of them are odd, 5, = —63 In both cases, 6, is nilpotent

(63 = 0) However, neither 5, nor 6) is nilpotent when there are simultaneously odd and even constraints

3.2 Main property of 5, and 6,

From now on, we will assume that the constraints are “irreducible,” or, as one also says,

“independent,” which means that the (super-) Jacobian matrix (0G,/dz,) contains a m X m submatrix

that can be inverted

This implies that one can find, at least locally, an invertible change of variables (not necessarily

canonical) such that the functions G, become, in the new coordinate system, the m first coordinates

n order to 1 ate the diff in demon

case when all canonical variables and all constrai

yrTainary h rÌ 112 ry A 2a rÌ (yr ( a ñ 1e OT) 11T] ‘Ta E ry 4 a A ( On

F = 8K, as one can easily show by purely algebraic means However, on the constraint surface, the

G ay -f} Amnlete dagenerate ang her 1c 2S giasantee ha 2^>2a1ine all hehavedr h a

Y /2 — {j) s 5 yu H d > o kÌ 2 Bs idl a d vy L) ay s

Now, we want to control K close to and on the constraint surface, where the system remains

10 Marc Henneaux Hamiltonian form of the path integral for theories with a gauge freedom

classically It 1s here that the irreducibility conditions play an important role (It is without saying that all the necessary regularity conditions on the functions F “'’“? (differentiability, .) are also assumed.) Coming back to the proof of the theorem and to the general case of commuting and anticommuting constraints, let us first note that the operator 6, which explicitly reads

in the coordinate system where the functions G, are the first coordinates y,, is linear in F and turns a

homogeneous polynomial of degree f (in y,) into a homogeneous polynomial of degree f+ 1

where the sum now bears over all permutations of (1, ,q+ 1) By construction, K, is completely

antisymmetric This proves the theorem, which indicates, incidentally, that it would be wrong to think that

62F = 0 implies F = 0

3.3 Equations involving 6, and 64

From the main property of 6, and 63, the following theorems are easily demonstrated

Theorem 3.3: A necessary and sufficient condition for the existence of solutions to the inhomogeneous

where E is a particular solution, and where K is arbitrary

The theorems remain true if one replaces everywhere 62 by 63, and ô; by 62

Since they are closely related to C*%.,, they will also be called the “first-order structure functions”

associated with the functions G,, which will be called themselves the zeroth order structure functions

By virtue of the Jacobi identity for the G,, which one can write as

12 Marc Henneaux, Hamiltonian form of the path integral for theories with a gauge freedom

(i)

as one can see by working out (4.1.3) using (2.11a) Here, we have set

Equations (4.1.4) have the form 6;D=0 As it follows from the previous section, they imply (2)

D = &U, or, more explicitly,

(2)

This relation defines the second-order structure functions U,,,,,," which are completely antisymmetric

in both (b;, bz, 63) and (a), a2) They can be chosen so that their Grassmann parity 1s

(2)

£(u"”) = &p, + Eb + Eb, + Ea, + Fa (4 1.7) The factor 2 in (4.1.6) is just a matter of convention

4.2 Third order structure functions

The _ third-order structure functions are introduced along similar lines Let_us take the Poisson

bracket of (4.1.6) with G,,(~)**«"") and antisymmetrize the resulting expression in the indices 4;

One gets WES SY

~“ 2) (1)

y

4, fo y aya r> (> _f— Ve bgt DG gs + DY ey đ1 —\Ceagt DG w+ D4 Ay \

ÚU= |2 hp; Gaz Gol) la TP rire) Gal) JA (4.2.1)

The first term on the right-hand side is obviously a combination of the constraints,

ebgt 1 Mea, +1 — 2 Ebat |Meat €at 1)

[2 Up, Gan, G,,(—) bat 1 Mea, Va — [4 Us, poy" U„„* (-)! bat eat

Besides the antisymmetrization in the indices (),, bo, bs, b,) there is, in the second term of the right-hand side of (4.2.4), an additional antisymmetrization in the indices (bs, by, k)

We now transform the last two terms of (4.2.4) by means of formulas (4.1.5) and (4.1.6) Using the

symmetry properties of the first-order structure functions and the antisymmetrization on 5,, 52, b3 and b,, one is left with

[D»u"', G„(—) 829642 ĐÌ, = ((—) 966832 99[ Um^t, Uyu,^]Œ¿„)A + 6[(—) #8" #41 Ug

If one substitutes (4.2.2) and (4.2.5) in the relation (4.2.1), one finally gets the identity (2)

This relation defines the third-order structure functlons, which are completely antisymmetric ¡in

(bạ, bạ, b3, b4) and (a,, az, a3) Because of theorem 3.4, the third-order structure functions are not

14 Marc Henneaux, Hamiltonian form of the path integral for theories with a gauge freedom

That something precisely defines the fourth-order structure functions The defining relation of the

fourth-order structure functions generates in turn new identities, which lead successively to the structure functions of order 5, 6, 7, along exactly the same lines

The structure functions of order (n+ 1)U St? ""! (n = I, 2, 3, ) are accordingly all defined by

‘structure equations” which take a form similar to (4.1.6) and (4.2.8),

Because of the way they are defined, the structure functions of order (n + L) contain some ambigulty

Indeed, as stated in theorem (3.4) one can add to them an antisymmetric tensor of order (n + 2) contracted with G, without changing (4.3.1) One partly removes the ambiguity by demanding that the

structure functions have the Grassmann parity

The coefficients D,, 5,,.°° °" appearing in eqs (4.3.1) contain the brackets of two U’s of order <n,

as well as the contractions U *U ,° (contraction on only one index) The only terms which fulfill

aan — 1 aap Gps an) f_\C2"* Jeb t nn cp+ L) CÔ? D1 cap tpt 1) 5S” Đi ta t1 H op

D2 2 [ s¡ „ ` U bps bas? | ( ) ¡ị pr^ “Ti 7 a pepe] fay f

is included in an appendix because it is rather cumbersome and not very illuminating (it just generalizes the steps of the previous paragraphs) Let us simply note here that the formula (4.3.4) reduces to (4.1.5)

alls 4 D more nan ne mere

s, together with their canonically expressions They will enable u

kỉ k1 €liti.YVY ti 11 11CCll¿ liiC apvivyp

conjug

ate momenta, to exhibit the

l6 Marc Henneaux, Hamiltonian form of the path integral for theories with a gauge freedom

canonical invariance of the theory under changes of the structure functions, which, as we have pointed out, are ambiguously defined They will also become the ghost variables of the path integral For that

reason, they will be called from now on “‘the ghosts.”’ (Their conjugate momenta F, will be defined in the

(1) (n)

next section and will, among other things, take care of the antisymmetrization of U and D in their

upper indices At this stage, though, 1.e., in the proof of (4.3.4)-(4.3.9) their introduction is not

for n Then, one finds, using (2.4d), that

It is clear that the rank of a set of constraints can be changed, since there is an ambiguity in the

definition of the structure functions, and one can raise or lower the rank by appropriate modifications of

these functions Hence, the concept of rank is only meaningful for a given set of constraints and

G, = 0 by equivalent constraints F, =0 which strongly commute ([F,, F,| =), so that the rank of a

theory can always be made equal to zero (However, the computation of these equivalent functions F, is hard in practice, and it is better not to assume that the constraints that one is given have identically vanishing brackets.)

The following theorem has been found useful in the determination of the rank of a system of structure functions

Theorem 4.2: If all structure functions of order k are zero, with s<k<2s + 1, then, all structure functions of order greater than 2s +1 also vanish and the set of structure functions is at most of rank s

n exceeds the number m of constraints because there is no antisymmetric tensor of order >.)

5 The generator (2 of the Becchi-Rouet-Stora (BRS) transformation

5.1 Extended phase space The extended phase space is defined by introducing m extra variables Y, canonically conjugate to

the ghosts 7°,

ghost number zero, the ghosts 7* have ghost number one, whereas their momenta Y, have ghost number minus one Moreover, one requires that the ghost number of a product of variables is equal to

the sum of their ghost numbers (ghost number (777°) = + 2, ghost number (n7,) = 0, etc.)

5.2 The generator 2

Let us consider the series

Is Marc Henneaux, Hamiltonian form of the path integral for theories with a gauge freedom

(n)

n=0

; (1) where the coefficients )“"”" are such that (2 has Grassmann parity plus one,

The first condition expresses that to zeroth order in the momenta Z„ the transformation generated by

(2 is just a gauge transformation in which the infinitesimal parameter e“ is replaced by the ghosts 7°

The second condition means that (2 is nilpotent, 1.e., that the second variation

Let us stress that 2 is odd, so that the nilpotency condition [Q, 2] = 0 is a non-empty requirement The properties (5.2.3) and (5.2.4) of 2 makes it very similar to the generator of the supersymmetry

transformation invenfed by Becchi, Rouet and Stora in the Yang-Mills case [10,11] It is thus

appropriate to call 2 the generator of the BRS transformation, even for more general constrained systems associated with open algebras

where the “''^* are Just the structure functions of order x

The proof is straightforward Let us compute explicitly [Q, Q] One finds, after some rearrangements

— _(")

(n+1) (n+1)

QQ a ananst ss [J 9U 7nềmH, (5.2.10b)

A ording ed (* 0)ì—t.e he nilpoten ondition 4)— leads, togetne vith the “‘initiz ondition”’

(n) (5.2.3), to the conclusion that the coefficients (2 *'"’* are just the structure functions associated with the

anoni nsformation in the extended phase space The ambiguity pla hus no critical role

For Abelian theories, the BRS generator reduces to

rn a b CON ON Std An ` L) O mone AT) d neorive OS 5 noweve L) we VOLS he generator ciliCL(đttLC Oo d ne

does not have that simple form and contains higher powers of the ghosts

S Ñ a

20 Marc Henneaux, Hamiltonian form of the path integral for theories with a gauge freedom

It is easy to check that the ghost number of {2 is equal to one, since there is always one more ghost than

there are ghost momenta Besides, 2 is real,

as it follows (i) from (4.3.10), (ii) from the relation

(Pant? Pay)* = Pays Pag(- Per (5.2.14)

(n)

and (iil) from the fact that U*'”"*" commutes with the product ¥, -,, since they are of opposite

Grassman parity

5.3 Comments

In this section we have proved the existence, for any system of first class constraints, of the generator

of the BRS transformation This result is purely classical and entirely follows from the classical structure

of constrained Hamiltonian systems We have also shown that the BRS generator only depends on the form of the gauge transformations (i.e., on the constraints), but not on the dynamics (i.e., on the Hamiltonian)

One can actually view the BRS generator as the “generating function” of the structure functions of the theory, since these functions appear as coefficients of 2 in its expansion in powers of the ghost

momenta In that picture, the nilpotency conditon [Q, Q] = 0, which plays a key role in the quantum

theory, appears as the “generating function” of the structure equations (4.3.8) which define the

In the Dirac theory of constraints, one sometimes call “observables” real even phase space functions

Ao(z) which have weakly vanishing Poisson brackets with the constraints [14, 26]

[Ao, Ga] = W,’G, = 0 (6.1.1)

“observable.”

Two observables Ao, Ag are considered as equivalent if they differ by a term which vanishes on the constraint surface,

In that case they possess the same classical values

It is easy to show, by using the Jacobi identity, that the set of observables is closed under the Poisson bracket operation Furthermore, it follows from the first class property of the constraints that if Ag and

Bo are respectively equivalent to Ao and Bo, then, their Poisson bracket [Ao, Bo] is equivalent to

[Ao, Bo] Accordingly, the space of equivalence classes of observables defined by (6.1.2) is naturally

equipped with a Poisson bracket structure Note that in the open case, one cannot find, in general, a representative in each equivalence class which obeys [Ao, Go] =0 (strongly), for these equations

generically possess the constant functions as their unique solutions

The purpose of this section is to demonstrate the existence of appropriate BRS invariant extensions

of any observable Apo in the extended phase space, i.e., of appropriate extensions A obeying

By appropriate extensions, we mean (i) that A is equal to Ao when the ghosts and their momenta are set

equal to zero; (ii) that A has ghost number zero; and (iii) that A is real and even

For definiteness, let us assume that Ao is the Hamiltonian Hy (but the result holds for any observable) The searched-for H can be expanded in powers of the ghost momenta 9%,

2? Marc Henneaux, Hamiltonian form of the path integral for theories with a gauge freedom

One can show that these conditions are identically fulfilled —-and, consequently, that H exists — by

using repeatedly the Jacobi identity and the first class assumption [Hy, G,] = V.” Gs The demonstration

Here, A is an arbitrary external function which has zero brackets with the canonical variables

(m) Theorem 6.1: One can choose the structure functions U“""4" with ø > 1 in such a way that (i) they do not depend on A;

(ii) they vanish whenever one of the upper indices A; is equal to zero

The proof goes again by induction First, the theorem is certainly true when n= 1 (see formula (6.2.3)) Let us assume then that it holds for a given ø and show that it follows for n+ 1

34 Marc Henneaux, Hamiltonian form of the path integral for theories with a gauge freedoni

(H}

where the last transformation results from the A-Iindependence of 2“"”“, This proves the theorem 6.1

The existence of the BRS invariant Hamiltonian now easily follows from the nilpotency of Q'(A) If

one expands (2'(A = 0) in powers of the anticommuting ghost 7°, one gets

where both @2 and H live in the original extended phase space (24, 7“, 4,) without the new ghosts 7",

Py The property [Q'(0), Q'(0)| = 0 reads explicitly

and hence

Since 2 starts like 7°G, whereas H reduces to Hy when the ghosts are set equal to zero, 2 and H are

indeed the BRS generator and the BRS invariant Hamiltonian, as it was anticipated in the notations

(n)

The coefficients H in (6.1.3a) are the “structure functions” characterizing Hy and, since H exists

they obey the equations (6.1.7) It is easy to check that the first-order “ghost correction to Ho is simply

given by the first-order structure functions (2.2b) contracted with A, and n”,

where “more” stands here for terms containing at least four ghosts

ì213na-nve CC (2y `} On LÍ On c na TT V¥ Ca 3 k} ry 1⁄22 cl z Fe @ 1 ` ep“ Ol cl ana ov 7 q ta = k7 Ul L7 Te q

first class system (Ho, G,) is sometimes called the rank of the Hamiltonian constrained theory based on

previous sections have reveared the—existence structure functions—assoctrated with viv

constraints and have indicated how these structure functions enable one to construct the generator 22 of the BRS transformation We show here that these results are closely related to the old theory of

function groups and surfaces in symplectic manifolds, as developed by Lie and others (see, e.g [25]

chapters 6, 65, 66) At the same time, we prove that two generators {2 corresponding to two different choices of the structure functions are related by a canonical transformation in the extended phase space

A system of functions F, of the canonical variables 1s said to be a function group if the brackets [F¿, F„] are all functions of the F, The function group is commutative in the particular case when the

brackets [F,, F,] vanish

Now, the constraint functions G, do not necessarily form a function group since the C“,, might not

be functions of the G, However, the equations G, = 0 can be replaced by equivalent equations F;, = 0)

such that the new functions F, are identically in involution, i.e., form a commutative function group

(LF, F] = 0) It will turn out that the existence of the structure functions and of 2 just reflect that

property

To show the existence of the functions F,, one proceeds as follows Because the constraints are

assumed to be irreducible, one can solve the equations G, = 0 for m of the coordinates and momenta (at least locally), say 4“ and p„»,

Here, z, stands for the “dependent” canonical variables q* and p,, whereas the phase space functions

82 do not involve Z,

The functions F, = Z, — 4, which vanish on the constraint surface by construction, are combinations

of the G, Hence, their Poisson brackets vanish weakly But these brackets cannot involve z, since

Za — Za IS linear in z, with constant coefficients Consequently, they must vanish identically Accord-

ingly, the F,’s provide a set of searched for commuting functions which define the same constraint surface

as the original G, ([65], theorem VII.24)

Incidentally, we note that this property means that two first class constraint surfaces with the same number of fermionic and bosonic irreducible generators G, are (locally) canonically isomorphic, 1.e., one can find a canonical transformation which maps one on the other

This result could have been also obtained by showing that the two-forms induced on these constraint

surfaces by the phase space symplectic structure have identical features Well-known theorems of symplectic geometry imply then that these surfaces are (locally) canonically isomorphic (for more on symplectic geometry, see e.g [1, 2])

commuting

ry he “x1 = A rm

L) O Cd uitad 111G WimCTđI DOtf CD = SN CD

UT On a ana _O@q, WH Ì vậađ

of that surface in the terminology of Schouten [65], chapter II), are related by

in the vicinity of the constraint surface

The above theorem asserts the existence of functions a,’ obeying

26 Marc Henneaux, Hamiltonian form of the path integral for theories with a gauge freedom

or explicitly

G.{[aa°, Gala’ —|a’ Gylag? + faa’, af \Ga + ag? a Cena} =0, (7.2.3b)

For this to be the case, one must have, according to the theorems of section 3

[da° Ga]a.“ — [a.° Gy]ag? + 3[au’ a“ |Gy — sfas ag’ |Gy + ag? a! Cena = Kad! G, (7.2.4)

for some functions K,./ = — Kad® =~ Kea‘

On the constraint surface (G, = 0), these differential equations for a,° reduce to

|aa° Gala! ~ [a.“ G,]a„" + a,” a! Cn = (7.2.5) and can be loosely interpreted as yielding the exterior derivative da of a on the submanifolds

canonically generated by the constraints under the Poisson bracket operation (i.e., the integral

manifolds generated on the constraint surface by the gauge transformations)

Now, eqs (7.2.4) will be integrable only if d*’a=0 Computing the bracket of (7.2.4) with G,

multiplying the answer by a“,, and adding to the result the similar expressions obtained by per-

mutations of the indices (a,c, m) (odd permutations being affected with a minus sign) one gets, after some rearrangements involving the Jacobi identity and (7.2.4), two types of terms:

(a) terms which are not multiplied by the constraints and which do not obviously vanish, even

weakly These read

where d is the differential operator defined as above (d involves antisymmetrized brackets of K,,° with

nd _essentia educes to the ordinary exterior derivative operator on the surfaces generated by the

constraints For that reason, d is loosely called below the exterior derivative operator)

integrability conditions of eq (7.2.4) must be fulfilled

DU W R UW (J s Ge ` ad ¿

weakly zero (eqs (4.1.5) and (4.1.6)), so that the integrability conditions of (7.2.4) are satisfied on the

constraint surface Hence we only have to worry about the second terms (7.2.7) Demanding that these

cancel the contribution from (7.2.6) containing the second-order structure functions —so as to fulfill the

here to terms which do not contain [K, G,] and whose explicit form does not play a key role at this level

of the analysis Equation (7.2.4) must be supplemented by (7.2.8)

One gets by repeating the procedure on (7.2.8) (i.e., stating that d’*K = 0) an equation involving this

time (dL) and some new functions M 2g", (antisymmetric in both sets of indices), whose “exterior

derivatives” are restricted in turn by the condition d*L = 0 If one keeps doing this, one finally gets a system of equations

(7.2.9d)

for the unknown functions a4, K%, L22, M@B,, Its explicit form will not interest us here

In order to find the functions F, along the above lines, one has to solve the system of equations

(7.2.9), As we have sketched — and as we know from the previous paragraph — this system is integrable:

taking the “exterior derivative” d of any of the equations (7.2.9) and using the identities obeyed by the

structure functions, one gets the next equation

Hence, the identities satisfied by the structure functions appear as the integrability conditions of the system (7.2.3a) ((7.2.9)), which we know possesses a solution The existence of the structure functions

is accordingly intimately connected with the possibility of replacing any set of first class constraints by constraints identically in involution

ust as one can (locally) replace the constraint functions G, by ions F, in involution, one can

define an equivalent Hamiltonian Hy which strongly commute with the F,,

which differs from Hp by a combination of the constraints (Hj — Ho vanishes on the constraint surface)

vanish identically _

ry +

ne Œ 2T -Ì Gj k7 “ k7 ‘yr ie are way oy = ae q TT k2 On s 1T] LỊ ne ⁄ "he

condition [Họ + À“ F¿, F,] = 0 implies the equations

28 Marc Henneaux, Hamiltonian form of the path integral for theories with a gauge freedom

on the unknown A‘ Here, u," are some functions antisymmetric in ac

The integrability conditions of (7.3.3) read

[V4 Fe] -[W% Fal = {Lae Fe] — lee Fe Hi (7.3.4)

The right-hand sides of (3.4) vanish on the constraint surface whereas the left-hand sides can be transformed as

(i.e., they are the coefficients of the ghosts in the H of section 6)

Hence, the integrability conditions of (7.3.3) hold as a result of the identities for the structure

functions provided y,* is a solution of

@)

[us Fo] — [ue FQ}— Vi = vie" F,, (7.3.6)

for some functions +” completely antisymmetric in (6, c) and (a, k,n)

a 1a | d ormatt O ] PTV ñStra 5 “a — (Te } a — Gen ‹

involution cannot be realized by a canonical transformation in the phase space of the original variables

It turns out, however, that one can transform the BRS generator Q“” = 4% G, + (appropriate higher

order terms in the ghosts) into the BRS generator 2°? = p” F„ by a canonical transformation in the

extended phase space This transformation explicitly involves the ghosts In that sense, one can say that

the passage from G, to F, 1s realized by a canonical transformation

That 2 is related to Q”%? through a canonical transformation is not unexpected, since these

generators are both nilpotent, (2°, 2] = [N72] = 0

We will construct the desired canonical transformation in the case when the constraints G, and F,

differ infinitesimally,

with ¢,” small The general case can be obtained by exponentiation

The question reduces then to that of finding the generator C of the canonical transformation such

that

where

(1)

NYO - IP = n°e%, F, t+ 9° 9% Ua P + Ole?) (7.4.3)

(the higher structure functions of the set G, can be taken to be of order e* or higher)

This generator C is obviously given by

It is even, has ghost number zero (which means that it leaves the ghost number invariant) and one

easily checks that it leads to the change (7.4.3) in 2 (We allow again for the possibility of anticommuting constraints and canonical variables.) Hence, 2%? and OQ are indeed related by a

generators £2, (2' associated with a given constraint surface are also canonically equivalent In other

words, the ambiguity in the structure functions of order =1 mentioned in section 4 can be completely

compensated for by a canonical transformation in the extended phase space

The direct proof goes as follows: suppose 2 and 2’ are identical up to order (p — 1) in the momenta Y,:

the first (p — 1) terms are the same, but the terms of pth order differ Consider the canonical transformation

This transformation does not modify the terms of order less than or equal to p— 1 However, it does

3U Marc Henneaux, Hamiltonian form of the path integral for theories with a gauge freedom

In particular, the change in the term of order p is given by

` a — ta} ap: ee Wtf adh» chy a Gln ` C}] — (~) Ga Ui a 1) 'C đ1 dyed Po , Pry, (7.4.6) (we assume again that the canonical transformation ts infinitesimal)

But the ambiguity in the pth order structure functions (for given structure functions of lower order) Is precisely of the form (7.4.6),

(Pp) (ps

bị: -bọ Pe Ua an hị - bạ — OP EGR ay taf? I’ ah hy (7.4.7)

where K yap is antisymmetric in both sets öŸ Indices (ai đ„.¡), (d b,) 1e., has the

same symmetry properties as Cy, 4,,,°"" " Hence, by an appropriate choice of the functions

Cacao?! in (7.4.5), one can make the terms of order p in both Q and 2’ equal I pri fb q

One can repeat the argument, this time for the terms of order p+ 1, and reach similar conclusions

Accordingly, one can make 2 and 2’ coincide to all orders by a suitable canonical transformation in the extended phase space

This result is of fundamental importance because it indicates that the ambiguity in the structure functions is harmless in the classical theory and simply leads to canonically related realizations of the BRS transformation One can thus say that with the introduction of the ghosts, the canonical covariance

of the structure of constrained systems becomes manifest

In the quantum theory, where 2 becomes an operator, different choices of the structure functions

lead formally to unitarily related quantum BRS generators [7,75] (we do not address the ordering

Again, it is clear that one cannot describe the passage from the original H, to the

h On a ne On saints p ‹ Anon ans 11a 1 h1 4 Dn‹ ye

transformation would have to change the Poisson brackets with F,

in generally covariant theories, and Hy = A“ F,, ~ 0 The function H associated with Ho does not vanish

Now, no canonical transformation can furn A; = 0 into a non-zero function

In order to determine how H and Hj, (and more generally, how two different BRS-invariant extensions of the Hamiltonian) are related, let us come back to the construction of H, which used two additional, canonically conjugate ghosts 7° and Ay

With these variables, H appears as the coefficient of 7" in the expansion of the BRS generator Q=2+n°H associated with the system of functions (G,.H,) But, according to the previous paragraph, two different 2,’ are related by a canonical transformation in the enlarged phase space

Let us expand the generator U of the canonical transformation which maps 2 on Q’ in powers of 7"

U=U+7K (7.5.1)

U has ghost number zero and is independent of n°, whereas K has ghost number minus one and is also

independent of 7° Because neither 2 nor 2' contains Po, one can assume that U, and hence U and K,

do not involve Y, either _ Let us evaluate 62 = [.2, U]:

The first term on the right-hand side of (7.5.3) was expected and represents the change of H due to a

canonical transformation in the original extended phase space

The second term is new and indicates that one must consider as equivalent two Hamiltonians H, H'

which differ by, a term like LK, 2), where K has ghost number minus one and is antlcommuting,

Pr Swit ai Heh are wo, a ˆ Ve c RAVE a S ki E CCH, H Suc 8 chanc a - | ry c1 aAY c“cOUún k2 a ne =_€ ambig kì in a `

functions associated | with Ao

constraints identically in involution, so that the rank of any theory is actually always zero

AAP ac nh A yr? 1 f^ € ¬w r1 yor 1Œ ROL a a D ˆ h Ð nÐs<e£

7) Ñ GU Oli WITTY WU y¥ adDOU Oy AIS CUI ait k7 k7 ĐT s > 7

theories only result from a “bad” choice of the constraint functions?

ll Marc Henneaux, Hamiltonian form of the path integral for theories with a gauge freedom

The answer to this question is at least threefold First of all, given constraints whose brackets only

vanish weakly, it may not be practically easy to find the equivalent constraints F, = 0 such that

[F¿ F,] = 0 One thus wants a quantization procedure which avoids having to get these F,, and this is precisely what ts reviewed here

Second of all, the existence of the functions F, is only guaranteed locally (in phase space) It is

accordingly desirable to use a method which does not explicitly use them

Finally, in the case of field theory, the passage from G, to F, is generally non-local in the spacetime

coordinates Again, one might not want to use non-local constraints F,, especially if one feels that there

is some fundamental character in the locality of the physical laws and of the action principle from which they derive In addition, the use of the F,, might spoi! manifest Lorentz invariance

Our last comment concerns the fact that the group concept has been evidently deemphasized in the present analysis, something we actually wanted since the aim was precisely to cover the open algebra

case In that respect, the geometrical interpretation of the ghosts as some Maurer—Cartan form on the

group of gauge transformations is of no immediate relevance to the above construction of (2 (Besides,

even when the constraint functions generate a group, that interpretation of the ghosts depends on how one represents the constraint surface, since two different groups can lead to equivalent constraints

Example: the constraints p, = 0, p> = (0) (Abelian group) are equivalent to the constraints (exp g2)p; = 0

p2 = 0 (non-Abelian group).)

Of course, our analysis so far 1s classical The group structure, when there is a natural one, plays, as Is

well known, an important role in the quantum theory (anomalies, )

8 Quantum theory — general aspects

with a non-positive inner product This inner product is such that the real canonical variables become

ne representation of the odd ghost opera n“ and 2 „ rn the space of functions of the Grassmann variables 7° has been discussed for instance in [41] It goes as follows when one assumes only one pair

One has

The scalar product is given by

from which one finds that 7 is Hermitian whereas is anti-Hermitian Besides, one easily gets

(,1)=0=(%,7), (U,9)=(, D*=1 (8.1.6)

The state 7 is sometimes denoted by |0) since it is annihilated by the operator 7 It has zero norm

We will assume throughout that one can order the BRS generator 2 in such a way that it is not only Hermitian but also nilpotent, [.2, Q], = 20? =0 This latter condition is the quantum expression of the classical property [2,.Q]=0 Note that these two requirements (2? = 0, 2* = 2) are compatible only

with an indefinite metric

Quantum observables are zero-ghost number operators A which commute with 0, 1.e., which are

BRS invariant

This definition is again motivated by the classical theory, where the property [A, Q| = 0 implies that the

zeroth order term Ao in the ghost variables has weakly vanishing brackets with the constraints, and

hence, is gauge invariant By virtue of the Jacobi identity for the commutators, the commutator of two

quantum observables is also_a quantum observable

The ghost number of an operator is defined as in the classical theory It can be represented by the

_ Aa

Aa

rom the relations (8.1.8), it follows that if a state as definite ghost number clf)= , then

f) and Y,|f) have ghost number k+1 and k-1,