Duality symmetric actions and canonical quantization

Physics Letters B 323 (1994) 311-315

North-Holland

PHYSICS LETTERS B

Duality symmetric actions and canonical quantization I. M a r t i n a n d A . R e s t u c c i a Departamento de Fisica, UniversidadSim6n Bolivar,Apartado89000, Caracas, Venezuela

Received 11 November 1993 Editor: M. Dine

We obtain an extension of the Schwarzand Sen formulation of dual symmetric actions, avoiding the second class constraints present in their approach. The new actions are also symmetric under dual transformations and present an infinite set of gauge

symmetries.

In the search for a better understanding of the full non perturbative string theory, recent research [ 1-3 ] has led to the study of discrete symmetries that might be present in the full string theory, the so called " T " and "S" duality symmetries. The reason for this conjecture lies in the fact that both symmetries are manifest at the low energy effective field theory limit, " T " duality in the action and "S" duality in the equations of motion. Lately, Schwarz and Sen [4 ] have devised a method for making "S" duality a symmetry of the action in the hope of attaining results similar to the ones already given by the " T " duality symmetry or "target space duality" in the string theory. In this paper, we make an analysis of the consequences for quantization, once "S" duality is a manifest symmetry of the action. The manifest local realization of this symmetry introduces the inherent difficulties associated to second class constraints in the canonical formalism for quantization. We begin by studying the canonical quantization of the "generalized" Maxwell action with a local duality symmetry, equivalent to the usual duality between electric and magnetic fields of the Maxwell equations of motion. This action was introduced by Schwarz and Sen to illustrate their method for elevating the duality symmetry to the action. In the process of quantization we find second class constraints that introduce off shell non-localities spoiling the procedure. In the "generalized" Maxwell case, these difficulties may be overcome, but that will not be the case in supergravity theory, the low energy field theory of string theory. We propose, ElsevierScience B.V. SSD1 0370-2693 ( 94 )00019-4

then, a more general class of Maxwell's actions with manifest local duality without second class constraints and with an infinite set of gauge symmetries. These will reduce to the one given by Schwarz and Sen, when a suitable gauge is fixed. In the following, we make an analysis of the duality invariant Maxwell action, proposed by Schwarz and Sen [ 1 ], using the canonical formalism. We will show that such an action will derive second class constraints which will introduce non localities in the process of quantization. Even when these non localities may be harmless in Maxwell's theory, they imply serious problems when gravity plays a role as in the low energy limit of superstring theory. The duality invariant Maxwell action is - ½< B ' ~ % # E :

+B~iB~'>,

where

E iO-L O o A i O~ - O i A o ,Ot or=l, 2,

BCa=cjkOjA$,

i = 1 , 2 , 3,

(1)

and < > denotes integration in four-dimensional Minkowskian space. It possesses a local duality symmetry under the transformation

A u1 ~ A u2 ,

AZu--,-A~u,

/l=O, 1,2, 3

(2)

which reduces on shell to the standard dual transformation

E~B,

B--,-E. 311

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PHYSICSLETTERSB

In terms of the canonical formalism on phase space, a canonical conjugate momenta set n~ will be associated to the fields A~'(x), and consequently the following constraints will be derived: i -= n#i + ½B'~i¢,~#= 0. ¢p#

(3)

Here, the A~ are Lagrange multipliers that implement only boundary conditions on phase space, unlike the field Ao in the standard Maxwell theory that implements a constraint on the full phase space. In fact in ( 1 ), we have

where the Z p are the gauge fixing functions associated to the first class constraints and ~o~r are the transverse projections of constraints i.e. E0k0N~. If we eliminate n~ andA~ in terms of A~ and n~, respectively using the equations 0 i ~ = 0, ~ r = 0 and ~ff = 0 with the gauge fixing condition Z 2= 0iA 2, the integration of A~ and n~ may be performed in (8), leaving finally

I ~ A] ~nil 6(Oirtil) det{OiTt],g 1} 6(x I )

(i

-- ½< OiAgE#aBC~i> =-½(O,(AgB"')E~>+½(AgO, Ba'~>

,

(4)

(5)

where

H = ½B~iB"i .

(6)

The constraints (3) are a mixture of first and second class constraints. Its algebra is given as follows:

where HMAXWELL=½rC]lr] + ½BliBli and rr] has been redefined by a factor of 2. This is exactly the canonical formulation of Maxwell's theory. In the above analysis it has been implicit that the fields are defined on a simply connected space, this condition is crucial to derive Maxwell's theory from the Schwarz and Sen action. It may be easily proved, that the canonical action as well as the Poisson brackets remain invariant under the duality transformations

A~ ~eaaAPi , /tc~ i ~ Ea#~zp, i

i

(7)

Bli--~2nl

f ~A

B"~-E

~rt 5(O~O~l)g(O~o~) det{O/p~, )~fl}(~(X 1 ) 6 ( X 2)

Xg(tpff)g(~E-r) det~/2{~o~r, ~ r }

312

(10)

i

These transformations leave the functional integral (8) invariant, this in turn implies that the dual symmetry is realized off-shell. Once we eliminate A ~ from the constraints (3) the duality transformation (10) reduces to

Since each ~o~ may be decomposed in a longitudinal and transverse part, it is straightforwardly shown from (7) that the longitudinal projections, i.e. 0A0~ are first class constraints while the transverse projections are second class ones. The equivalence between ( 1 ) and Maxwell action was shown by Schwarz and Sen at the classical level, however the equivalence is also valid at the quantum level. In fact, the functional integral associated to (5) is

X exP(h < ~ t ~ O o A , - H > ) ,

(9)

where, in particular, the constraints 9~0 are transformed as follows:

{¢; (x), ¢i (x')} =o, {~(x), ~(x")} =0, {(o/,(x), q~ (X') }=ckJOk6(X--X ') .

)

X exp ~ < 7r~0oA 1 _ HMAXWELL> ,

where the second term on the RHS is identically zero and the first term is a total spatial derivative. The canonical action corresponding to ( 1 ) is < zr~0oA~-U+ft~(0~ > ,

17 March 1994

Itil--~-½B li ,

(I1)

further, using the Maxwell field equation 2z~ =0oA) - 0,A~,

(12)

eq. (11 ) becomes the standard duality transformation for the Maxwell fields

1 E1--,B 'i .

(13)

This is a non-local transformation for the potential A ~. In fact, eqs. (13) are equivalent to (8)

OoA~- O,A~-~OoXI -- 0iAo -1 =~ okO j A k1, ~ijka • 1 _ _ . i j k a ~ 1

ujZa k ~ t

1 + OiA 1

vj. * i = -- OoA i

(14)

the fields A~ are given in terms of a non-local expression of A ~ involving the inverse of a Laplacian. Notice that the consistency of (14) arises from the field equations for the gauge potential. We may notice that if in (9) X1 is taken to be Xl=OiA)

,

17 March 1994

PHYSICS LETTERSB

Volume 323, number 3,4

(15)

and the longitudinal parts of A) and n] are integrated out, the resulting canonical formulation for Maxwell theory is invariant under the non-local due transformation ( 11 ). In fact, the dual symmetry may be shown to be a canonical transformation on the transverse modes phase space. This result stands in contrast to the usual Lagrangian formulation of Maxwell's theory, where there is no invariance under the same symmetry. The Lagrangian formulation is obtained from the canonical one after integrating the transverse momenta n~r. It is equivalent to solving half of the canonical field equations and replacing the value of n ~ back into the action. Since this subsystem is not invariant under the dual transformation the procedure breaks the original invariance of the action even when the field equations continue to be invariant. The improvement obtained by considering ( 1 ), or its canonical counterpart, is that the dual symmetry is locally realized, while in the usual canonical Maxwell theory the symmetry is non-local. Nevertheless, the action ( 1 ) includes second class constraints that require further considerations in order to obtain a full quantum effective action. This is so since those constraints will require, in the standard Dirac quantization, the use of Dirac brackets, which in field theory will introduce non localities. In the case of Maxwell's theory, these non-localities may not be harmful but they present a serious problem for supergravity where we are interested in implementing, following Schwarz and Sen, something analogous to (1), supergravity being a low energy limit of superstring theory. To avoid the introduction of second class constraints we are going to construct a new action with more gauge symmetries than ( 1 ) and with a local realization of duality without second class constraints. Further, it reduces off-shell to (8), once we partially gauge fix it as a quantum effective action.

There is a standard way [ 5 ] of constructing an action with first class constraints that reduces to one with second class constraints, having Poisson brackets with the structure of (7), after a partial gauge fixing. However, we show below that this procedure, even when it leads to interesting results, breaks the original local duality invariance of (5). In the case of ( 1 ), there are two actions that could be constructed following the above mentioned procedure and that reduce to (8). We describe one of them and give the result for the other one. We consider the following set of first class constraints: g~ _=zt~ - ½ B 2 ' = 0 ,

~v2- 0,n~ = 0 ,

(16a, b)

their Poisson brackets yield zero. The longitudinal projections of ~v~ 0ig/~ = 0in{ were one of the first class constraints in the original formulation while (16b) was the other one. The transverse part of (16a), however, was a second class constraint in (3) but, in the set ( 16 ), is a first class one. This new first class constraint generates a new gauge symmetry which may be fixed by considering the transverse ofx~ + ½B1~, i.e. eOkOj( nk2 + 1B~k ) .

(17)

With this partial gauge fixing, the set of restrictions reduces to the set (3). Next, we will construct a new hamiltonian/-Tt satisfying the following relations, not satisfied by H in

(6): {& ~}= E v~,,

(18)

the Hamiltonian/t will reduce to H in the gauge ( 17 ). The relation (18) is a necessary condition for a Hamiltonian of a dynamical system with first class constraints. It will ensure that the functional integral of the new considered system with canonical effective action ( r ~ - n + 2 1 g/{ +22~z + a(CX) )

(19)

will reduce to the original effective action (8) once it is gauge fixed by (17). In (19) the 2's are Lagrange multipliers, the C antighost fields and the ~ the BRST transformation associated to the BRST charge Q - C i ¢/a" i After some calculations, we obtain/~ given by _

a

313

Volume 323, number 3,4

PHYSICSLETTERSB

ffI=H-BU( Tti2+ ½B li) _~ ½(~i .31_½Bli) (hi2 ~_ ½Bli) ,

(20)

it satisfies (18) with V= 0. The action (19) has first class constraints only and reduces to (5) after gauge fixing it. In the same way, we may construct a second action with the already mentioned properties. We consider the fist class constraints ~---~r~ + ½ B " = 0 ,

~,= 0,,~ = 0 ,

(21)

and the canonical effective action ( h A - H + q~2 + q l ~ + ~(CZ) ) ,

(22)

17 March 1994

where ql, ~, are a conjugate pair of new coordinates of the extended infinite dimensional phase space, cop,, is a c-number to be determined shortly. The transformation is performed by considering the new set of irreducible constraints

O,~(x) = 0 ,

(26a)

~ p - ~0~(x) + q~~p(x) = 0 ,

(26b)

• ~=0,

(26c)

where the superscript T denotes as previously the transverse projection. O,~(x) and ~0~p ~~ are first class provided

oJ~p(x, x') = ½~p~"%~(x, x'),

where the t/are Lagrange multipliers associated to the ~, and

and • is defined by

/7=H+B2'(n~ - ½B2')

• ilp(x)-qilp(X)-(wi~(x,x')~(x'))x,.

+ ½(n] - ½B2') (~r] - ½BZ').

(23)

(19) and (22) transform one into the other under the dual transformation (10). We have thus generalized ( 1 ) obtaining a pair of actions with first class constraints only, each of them equivalent as a quantum theory to the Schwarz and Sen action ( 1 ). These actions are not invariant under the local dual transformation (10) but transform one into the other. These extensions of ( 1 ) may be of relevance in the analysis of the SL(2, Z) realizations for the low energy heterotic string theories. However, in here we will be more concerned with extensions of ( I ) that are invariant under a local dual transformation. To do this, we will require an infinite set of auxiliary fields that will be arranged into generating functions in order to give an unambiguous treatment of the problem. First, we consider the following set of definitions and constraints. The original set of constraints rearranged in a reducible way are

O~,(x) - O~r~(x) = 0 , ~(x)-zr~+½B"i(x)e,,~=O,

We construct now an extension/t of the hamiltonian (6) satisfying {/t, first class} = ~] Vfirst class, to have a conserved set of constraints. The result is

n=½B~JB~J-e#~B'~#+½(q~

+~0~2).

(28)

It can be shown that the dynamical system defined by (26) and (28) is equivalent to (1) off shell. The functional integral constructed from the effective action associated to (26), (28) reduces to (8) after gauge fixing and integration of the auxiliary fields. To avoid the second class constraints (26c) we will need to introduce more new auxiliary fields, obtaining the following first class constraints:

O~(x) =0, ¢~,p(x) =0, O~p(x) -- • ~p(x) + ~'~p(x) = O,

(24a) a, f l = l , 2,

(24b)

we then transform the reducible second class constraints (24b) into irreducible first class ones by introducing new auxiliary field q~~ , as in ref. [ 6 ], defined by

¢,~(x)=-rfi~(x)+(o~'L,(x,x')~.(x'))x,, 314

(27)

(25)

(klp(x) = ~ ~-~ (x) + q~~B(x) = 0 ,

(29a)

and the second class constraints • ~(x) =0,

(29b)

where q~]p, ¢ ~ , q) ~p are the auxiliary fields which in terms of the independent canonical conjugate pairs are given by

Volume 323, number 3,4

PHYSICS LETTERS B

¢%(x)-n~ap(X) + ( O ~ ( X , X ' ) ~ ( X ' ) )x' , • ~a(x) = t/~a(x) - ( c o ~ ( x , x ' ) ~ ( x ' )

(30)

)x,.

The Hamiltonian o f the system is still given b y / 7 since it commutes with the new first class constraints. It may be proven that the new level n dynamical system, as we call it, is again off shell equivalent to ( 1 ). The functional integral constructed from the effective action associated to (2 8 ) and (29) reduces after gauge fixing and integration o f the auxiliary fields to the Schwarz and Sen formulation. The gauge fixing conditions are (~fl)T

.....

(~ifl)T=o

'

(~ti) L .....

(~noti)L = 0 ,

(31) (29) and (31) yield ~i ..... ~.i=0,(32)

r/~=...=t/.~=O,

(32)

(29) reduces then to the Schwarz and Sen constraints. Moreover the measure in the functional integral reduces exactly to the one associated to the Schwarz and Sen formulation. (28), (29) are invariant under the transformation A1---,, A 2

,

1 2 ~ai-'~ai,

A E --* -

2

A 1

1

~ai "-'~--~ai

,

7tl--~7~2 ,

7t2---~--Tt I ,

~a2-'~--l~al, (33) which extends the local duality transformation o f Schwarz and Sen to the full phase space. We finally consider the n ~ oo limit. That is, we define the following dynamical system. The Hamiltonian which determines the evolution o f the system is given by (28). The system is subjected to the first class constraints (29a), in the limit n ~ , there are no second class constraints. We now show that by gauge fixing we reduce the system to (29a), (29b). In fact consider the partial gauge fixing conditions (~ i~8)T=0

,

~]al--~a2,

,(33)

(~i)L=o

,

m>n,

(34)

associated to the first class constraints ~~~ m a , m > n. The first class constraints imply that ((~ ifl)L=

((~ ifl)L=O

.

17 March 1994

also ~ m p - - 0 , m > n . We have thus shown ~rni ~ 0 ,

i =0, t/m,,

m> n .

(36)

Moreover ~n+ l~ reduces to (29b) and is now a second class constraint since its gauge fixing was performed in (34). The action associated to the n ~ limit is then given by i "a i "or ( 7Cota i dl-~act~a i -n-~-gaOot dl-aal~iafl )

(37)

and, in the n - - , ~ limit, it is invariant under the local duality transformation (33). We have thus extended the original formulation o f Schwarz and Sen, introducing an infinite set o f auxiliary fields, to a new formulation with first class constraints only and with manifest local duality invariance in the sense o f Schwarz and Sen. The approach, which was used in ref. [ 6 ], is general enough to allow an analogous treatment for the low energy approximation o f the superstring. We will consider this problem in a following work. We should remark that the equivalence between Schwarz and Sen and Maxwell theory is strictly valid only on a simply connected phase space. The analysis of the equivalence in a multiconnected background requires additionally the introduction o f the cohomology o f closed forms. We have not dealt with this problem here, since the formulation is of course on a Minkowskian background. [ 1] A. Font, L. Ibafiez, D. Lust and F. Quevedo, Phys. Lett. B 249 (1990) 35; S.J. Rey, Phys. Rev. D 43 ( 1991 ) 526; A. Shapere, S. Trivedi and F. Wilczek, Mod. Phys. Lett. A 6 (1991) 2677. [2 ] J. Schwarz, preprint CALT-68-1815; A. Sen, preprint TIFR-TH-92-41. [3] I. Martin and A. Restuccia, Phys. Lett. B 271 (1991) 277; Anales de Ftsica, Monograflas. Vol. II 249, CIEMAT/RSEF (Madrid, 1993). [4] J.H. Sehwarz and A. Sen, preprint CALT-68-1863, CALT68-1866. [ 5 ] I. Batalin and E. Fradldn, Nuel. Phys. B 279 (1987) 514; I. Batalin, E. Fradkin and T. Fradkina, Nucl. Phys. B 314 (1989) 158; R. Gianvittorio, A. Restuccia and J. Stephany, Mod. Phys. Lett. A 6 ( 1991 ) 2121. [ 6 ] A. Restuccia and J. Stephany, Phys. Rev. D 47 ( 1993) 3437.

- - 0 and from We then conclude that ~ irap-

~~i, , # -__ 0

,

re>n,

315