aa r X i v : . [ h e p - t h ] J un Non-Local Gravity from Hamiltonian Point ofView
Josef Klusoˇn Department of Theoretical Physics and AstrophysicsFaculty of Science, Masaryk UniversityKotl´aˇrsk´a 2, 611 37, BrnoCzech Republic
Abstract
This short note is devoted to the canonical analysis of the non-local the-ories of gravity. We find their Hamiltonian and determine the algebra ofconstraints. We perform this analysis for non-local theories of gravity formu-lated both in Jordan and Einstein frame. The result of our analysis suggeststhat Hamiltonian formulation does not bring to clear identification of ghostspresence in non-local gravity. E-mail: [email protected]
Introduction
Recent experimental data suggests that the expansion of the universe is accelerating[1, 2]. One of the most popular approach how to explain current expansion of theuniverse is the introduction of cosmological constant dark energy in the frameworkof general relativity . Another possibility how to explain the acceleration of theuniverse is to modify of gravity action. The most well known example such a theoryare F ( R ) theories of gravity where R is the scalar curvature of D + 1 dimensionalspace-time and F is an arbitrary function, for review of F ( R ) gravity, see [5, 6, 7, 8].Another example of modifications of gravity that could explain the current accel-eration [9] are non-local modifications of gravity. This possibility is closely relatedto the proposal presented in [21] where authors suggested that the cosmologicalconstant problem could be solved in the context of non-local gravity. This ideawas further elaborated in recent papers [22, 23]. There are also additional reasonswhy it is interesting to study the non-local modification of gravity. For example,non-local effective field theories naturally emerge in the framework of string fieldtheory [10, 11, 12, 13, 14, 15, 16] and hence the string theories could provide naturalUV completion of non-local theories. For further analysis of non-local gravity fromdifferent points of view, see [17, 18, 19, 20].In summary, non-local gravity models are very intensive studied and deserveto be investigated further from different point of views. For example, one wouldlike to see how the non-local character of given theory is reflected in its canonicalformulation. The goal of this paper is to perform the Hamiltonian analysis of thebroad class of non-local theories of gravity [19]. We analyze these theories in Jordanframe and then in Einstein frame. We determine the constraint structure of giventheories and we argue that they obey the standard rules of geometrodynamics [25,26, 27] which is in agreement with the fact that these theories are invariant underdiffeomorphism transformations. On the other hand we show that the Hamiltonianstructure of given theories depends on the character of the non-local action. Moreprecisely, due to the fact that these actions contain derivative of scalar curvature itis convenient to introduce the appropriate number of scalar fields [19] and rewritethese non-local theories of gravity to the specific form of the scalar tensor theories.Then the crucial point is whether the scalar field A possesses canonical conjugatemomenta or not. More precisely, for the action where A appears linearly but whichis general function of (cid:3) − A, (cid:3) − A, . . . we find that this theory possesses collectionof two second class constraints. The presence of these constraints imply that thePoisson brackets between canonical variables should be replaced with correspondingDirac brackets. We also explicitly show that these Dirac brackets depend on phasespace variables. This is very non-trivial result whose origin can be traced to the non-local character of the theory. On the other hand we show that the Dirac algebraof the constraints takes again the familiar form and obeys the standard rules ofgeometrodynamics.We should also stress one important point. The present Hamiltonian analysis For review, see [3] and the most recent [4].
1s not sensitive to the fact whether some of the scalars have the kinetic term withnegative sign and hence should be considered as ghosts. This is a consequence of thefact that the Hamiltonian is linear combination of the first class constraints thataccording to the basic principles of the theory of constraints systems [24] shouldvanish on the constraint surface.Let us outline our results. We perform the Hamiltonian analysis of non-localtheories of gravity and determine their constraints structure. We find that theconstraints obey the standard rules of geometrodynamics. We also determine thecorresponding Dirac brackets between canonical variables for particular form of thenon-local gravity action. We derive equivalent results when we consider theoriesformulated both in Jordan and in Einstein frame.The structure of this paper is as follows. In the next section (2) we introducenon-local theories of gravity and map them to their Jordan frame. Then we per-form their Hamiltonian analysis and discuss the canonical structure of given theorywith dependence on the properties of the scalar field A . In section (3) we analyzethese theories formulated in Einstein frame and find their canonical structure anddetermine the results that are equivalent to the ones derived in (2) . We begin with the action for non-local gravity that was recently studied in [22, 23] S = Z d d x p − ˆ g (cid:20) κ d ) R (ˆ g )(1 + f ( (cid:3) − d ) R (ˆ g ))) − (cid:21) (1)where f is any function, ( d ) R is d ≡ D + 1-dimensional scalar curvature, (cid:3) isd’Alambertian (cid:3) = ˆ g µν ˆ ∇ µ ˆ ∇ ν = √− ˆ g ∂ µ [ √− ˆ g ˆ g µν ∂ ν ], (cid:3) − is the inverse of givenoperator and Λ is a cosmological constant . Due to the presence of the operator (cid:3) − it is convenient to introduce two scalar fields ψ and ξ and rewrite the action(1) in the following form S = Z d D +1 x √− g (cid:20) κ (cid:0) ( d ) R (1 + f ( ψ ) − ξ ) − g µν ∂ µ ξ∂ ν ψ − (cid:1)(cid:21) . (2)It is easy task to show that the actions (2) and (1) are equivalent. In fact, thevariation of the action (2) with respect to ξ gives (cid:3) ψ = ( d ) R (3) We use units of ~ = c = 1 and denote the gravitational constant 8 πG by κ = πM pl with thePlanck mass of M pl = G − / = 1 . × GeV . For simplicity we restrict ourselves to the analysis of pure non-local theory keeping in mindthat it is straightforward to generalize our analysis to the case of when the matter contribution ispresent. ψ = (cid:3) − d ) R . Then substituting this result into (2) we obtain (1).In what follows we will be more general and consider following general form ofnon-local action [20] S = Z d D +1 x p − ˆ g (cid:26) κ F ( ( d ) R, (cid:3) ( d ) R, (cid:3) d ) R, . . . , (cid:3) m ( d ) R, (cid:3) − d ) R, (cid:3) − d ) R, . . . , (cid:3) − n ( d ) R ) (cid:27) . (4)As usual it is convenient to map given action to more tractable form. Following [20]we firstly introduce scalar fields A, B and rewrite the action (4) into the form S = Z d D +1 x p − ˆ g (cid:26) κ F ( A, (cid:3) A, (cid:3) A, . . . , (cid:3) m A, (cid:3) − A, (cid:3) − A, . . . , (cid:3) − n A ) + B ( ( d ) R − A ) (cid:27) . (5)As the next step we define two scalar fields ξ , ψ in order to eliminate (cid:3) − A . Todo this we add following term to the action Z d D +1 x p − ˆ gξ ( A − (cid:3) ψ ) = Z d D +1 x p − ˆ g (ˆ g µν ∇ µ ξ ∇ ν ψ + ξ A ) . (6)At the same time we introduce two fields χ , η in order to eliminate (cid:3) A and addfollowing term to the action Z d D +1 p − ˆ gχ ( η − (cid:3) A ) = Z d D +1 p − ˆ g (ˆ g µν ∂ µ χ ∂ ν A + ξ η ) (7)so that the action (5) takes the form S = Z d D +1 x p − ˆ g (cid:26) κ F ( A, η , (cid:3) η , . . . , (cid:3) m − η , ψ , (cid:3) − ψ, . . . , (cid:3) − n +1 ψ )++ B ( ( d ) R − A ) + (ˆ g µν ∂ µ χ ∂ ν A + χ η ) + (ˆ g µν ∂ µ ξ ∂ ν ψ + ξ A ) (cid:9) . (8)From this analysis it is clear how to proceed further. We introduce following contentof the scalar fields A, B, χ k , η k , k = 1 , , . . . , m and ξ l , ψ l , l = 1 , , . . . , n . Thenrepeating the procedure presented above we can rewrite the action (4) into the form S = Z d D +1 x p − ˆ g (cid:26) κ F ( A, η , η , . . . , η m , ψ , ψ , . . . , ψ n )++ B ( ( d ) R − A ) + ˆ g µν ∂ µ χ ∂ ν A + ˆ g µν m X l =2 ∂ µ χ l ∂ ν η l − + m X l =1 χ l η l ++ˆ g µν n X l =1 ∂ µ ξ l ∂ ν ψ l + ξ A + n X l =2 ξ l ψ l − ) . (9)3his form of the action is our starting point for the Hamiltonian analysis of non-localtheories of gravity.As usual in order to formulate the Hamiltonian analysis of theory coupled togravity we have to introduce D + 1 formalism. Explicitly, let us consider D + 1dimensional manifold M with the coordinates x µ , µ = 0 , . . . , D and where x µ =( t, x ) , x = ( x , . . . , x D ). We presume that this space-time is endowed with themetric ˆ g µν ( x ρ ) with signature ( − , + , . . . , +). Suppose that M can be foliated by afamily of space-like surfaces Σ t defined by t = x . Let g ij , i, j = 1 , . . . , D denotesthe metric on Σ t with inverse g ij so that g ij g jk = δ ki . We further introduce theoperator ∇ i that is covariant derivative defined with the metric g ij . We introduce thefuture-pointing unit normal vector n µ to the surface Σ t . In ADM variables we have n = p − ˆ g , n i = − ˆ g i / p − ˆ g . We also define the lapse function N = 1 / p − ˆ g and the shift function N i = − ˆ g i / ˆ g . In terms of these variables we write thecomponents of the metric ˆ g µν asˆ g = − N + N i g ij N j , ˆ g i = N i , ˆ g ij = g ij , ˆ g = − N , ˆ g i = N i N , ˆ g ij = g ij − N i N j N . (10)Then it is easy to see that p − det ˆ g = N p det g . (11)We further define the extrinsic curvature K ij = 12 N ( ∂ t g ij − ∇ i N j − ∇ j N i ) , (12)where ∇ i is the covariant derivative calculated using the metric g ij . It is wellknown that the components of the Riemann tensor can be written in terms of ADMvariables. For example, in case of Riemann curvature we have ( d ) R = K ij K ij − K + R + 2 √− ˆ g ∂ µ ( p − ˆ gn µ K ) − √ gN ∂ i ( √ gg ij ∂ j N ) , (13)where K = K ij g ji and where R is Riemann curvature calculated using the metric g ij . Note that n µ has components n = 1 N , n i = − N i N . (14)4mplementing D + 1 formalism in the action (9) we find that it has the form S = Z d D +1 xN √ g (cid:26) κ F ( A, η , η , . . . , η m , ψ , ψ , . . . , ψ n )++ B ( K ij K ij − K + R − A ) − ∇ n BK − √ g ∂ j ( √ gg ij ∂ j B ) −− ∇ n χ ∇ n A + g ij ∂ i χ ∂ j A + m X l =2 ( −∇ n χ ∇ n η l − + g ij ∂ i χ l ∂ j η l − ) + m X l =1 χ l η l ++ n X l =1 ( −∇ n ξ l ∇ n ψ l + g ij ∂ i ξ l ∂ j ψ l ) + ξ A + n X l =2 ξ l ψ l − ) . (15)Using the form of the action (15) we can proceed to the Hamiltonian formalism.Explicitly, from (15) we determine conjugate momenta π N ≈ , π i ≈ , π ij = B √ g ( K ij − g ij K ) − √ g ∇ n Bg ij ,p B = − √ gK , p A = −√ g ∇ n χ ,p χ l = −√ g ∇ n η l − , p η l − = −√ g ∇ n χ l , l = 2 , . . . , m ,p ξ k = −√ g ∇ n ψ k , p ψ k = −√ g ∇ n ξ k , k = 1 , . . . , n . (16)Then after some algebra we find the Hamiltonian in the form H = Z d D x ( N H T + N i H i ) , (17)where H T = 1 √ gB π ij g ik g il π kl − √ gBD π − πp B √ gD ++ B √ gD ( D − p B − √ gBR + 2 ∂ i [ √ gg ij ∂ j B ] −− √ g p A p χ − √ g m X l =2 p χ l p η l − − √ g n X k =1 p ξ k p ψ k −− √ g κ F ( A, η , η , . . . , η m , ψ , ψ , . . . , ψ n ) + √ gBA + 2 ∂ j [ √ gg ij ∂ j B ] −− √ gg ij ∂ i χ ∂ j A − √ gg ij m X l =2 ∂ i χ l ∂ j η l − − √ g m X l =1 χ l η l −− √ gg ij n X l =1 ∂ i ξ l ∂ j ψ l − √ gξ A − √ g n X l =2 ξ l ψ l − , H i = − g ik ∇ l π kl + p A ∂ i A + p B ∂ i B + m X l =1 p χ l ∂ i χ l + m X l =2 p η l ∂ i η l + n X k =1 ( p ξ k ∂ i ξ k + p ψ k ∂ i ψ k ) , (18)5nd where π = π ij g ji . As usual the requirement of the preservation of the primaryconstraints π N ≈ , π i ≈ H T ≈ , H i ≈ . (19)As the next step we have to check the consistency of the secondary constraints withthe time development of the system. For that reason it is convenient to introducethe smeared form of these constraints T T ( N ) = Z d D x N ( x ) H T ( x ) , T S ( N i ) = Z d D x N i ( x ) H i ( x ) . (20)Then using the canonical Poisson brackets (cid:8) g ij ( x ) , π kl ( y ) (cid:9) = 12 ( δ ki δ lj + δ li δ kj ) δ ( x − y ) , { A ( x ) , p A ( y ) } = δ ( x − y ) , { B ( x ) , p B ( y ) } = δ ( x − y ) , { χ l ( x ) , p χ k ( y ) } = δ lk δ ( x − y ) , { η l ( x ) , p η k ( y ) } = δ lk δ ( x − y ) , { ξ l ( x ) , p ξ k ( y ) } = δ lk δ ( x − y ) , { ψ l ( x ) , p ψ k ( y ) } = δ lk δ ( x − y ) (21)we easily determine the well known algebra of constraints [25, 26, 27] { T T ( N ) , T T ( M ) } = T S ( N ∂ i M − M ∂ i N ) , (cid:8) T S ( M i ) , T T ( N ) (cid:9) = T T ( M i ∂ i N ) , (cid:8) T S ( M i ) , T S ( N j ) (cid:9) = T S ( N i ∂ i M j − M i ∂ i N j ) . (22)In other words the constraints (20) are preserved during the time evolution of thesystem. Note also that these constrains have to vanish weakly. As a result theHamiltonian has to vanish on the constraint surface and hence any instability relatedto the presence of the ghosts (which is general property of any non-local theory) isnot seen on the level of classical Hamiltonian analysis.It is important to stress that during the analysis performed above we implicitlypresumed that there is a momentum conjugate to A . However it turns out that forthe non-local actions that do not depend on (cid:3) ( d ) R the momentum conjugate to A is absent. More precisely, let us consider the action S = 12 κ Z d D +1 x p − ˆ gF ( ( d ) R, (cid:3) − d ) R, (cid:3) − d ) R, . . . , (cid:3) − n ( d ) R ) (23)that is the generalization of the action (2). Performing the same analysis as above6e find that this action takes the form S = Z d D +1 x √ gN (cid:26) κ F ( A, ψ , ψ , . . . , ψ n )++ B ( K ij K ij − K + R − A ) − ∇ n BK − √ g ∂ j ( √ gg ij ∂ j B )++ n X l =1 ( −∇ n ξ l ∇ n ψ l + g ij ∂ i ξ l ∂ j ψ l ) + ξ A + n X l =2 ξ l ψ l − ) . (24)From the action (24) we find the conjugate momenta π N ≈ , π i ≈ , π ij = B √ g ( K ij − g ij K ) − √ g ∇ n Bg ij ,p B = − √ gK , p A ≈ ,p ξ k = −√ g ∇ n ψ k , p ψ k = −√ g ∇ n ξ k , l = 1 , . . . , n (25)and the Hamiltonian H = Z d D x ( N H T + N i H i + v A p A ) , (26)where H T = 1 √ gB π ij g ik g il π kl − √ gBD π − πp B √ gD ++ B √ gD ( D − p B − √ gBR + 2 ∂ i [ √ gg ij ∂ j B ] − √ g n X k =1 p ξ k p ψ k −− √ g κ F ( A, ψ , ψ , . . . , ψ n ) + √ gBA −− √ g n X l =1 g ij ∂ i ξ l ∂ j ψ l − √ gξ A − √ g n X l =2 ξ l ψ l − , H i = − g ik ∇ l π kl + p A ∂ i A + p B ∂ i B + n X k =1 ( p ξ k ∂ i ξ k + p ψ k ∂ i ψ k ) . (27)We again introduce the smeared constraints T T ( N ) , T S ( N i ) and we easily find thatthey obey the relations (22). On the other hand the requirement of the preservationof the primary constraint p A ≈ ∂ t p A = { p A , H } ≈ N √ g (cid:18) κ dFdA − B + ξ (cid:19) ≡ N G A ≈ . (28)7inally we determine time evolution of the constraint G A ∂ t G A = { G A , H } == N − κ n X l =1 d FdAdψ l p ξ l − p ψ + πD − B D ( D − p B ! + 12 κ d FdA v A = 0 . (29)We observe that there are two possible alternatives. The first one corresponds to thesituation when d Fd A = 0 and we see that the equation (52) uniquely fixes the valueof the Lagrange multiplier v A . Then we can finish the analysis of the consistencyof constraints with the time evolution of the system since now p A ≈ , G A ≈ { p A ( x ) , G A ( y ) } = − κ d Fd A ( x ) δ ( x − y ) . (30)In principle these constraints can be solved for p A and A and hence we find theorythat has the same physical content as the F ( R ) theory of gravity coupled withthe collection of the scalar fields. It is also easy to see that the Dirac bracketsof canonical variables that define reduced phase space coincide with the Poissonbrackets.The more interesting example corresponds to the second situation when d Fd A = 0so that F has linear dependence on AF = AU ( ψ , . . . , ψ n ) + U ( ψ , . . . , ψ n ) (31)and hence the constraint G A has explicit form G A = √ g (cid:18) κ U − B + ξ (cid:19) ≈ . (32)Then the equation (29) implies an additional constraint G IIA = − κ n X l =1 dU dψ l p ξ l − p ψ + πD − B D ( D − p B ≈ . (33)Note that the Poisson bracket between G A and G IIA is equal to (cid:8) G A ( x ) , G IIA ( y ) (cid:9) = √ g (cid:18) − κ dU dψ + D − D B (cid:19) δ ( x − y ) ≡ △ ( x ) δ ( x − y ) . (34)The next step is to explicitly solve the constraints p A ≈ G A ≈ , G IIA ≈ p A ≈ A = const . As a result the pair A, p A is eliminated from the theory. Onthe other hand we solve the constraint G A for B and we find B = 12 κ U + ξ . (35)8n the same way we solve the constraint G IIA for p B with the result p B = 2 D ( D −
1) 1 κ U + ξ πD − κ n X l =1 dU dψ l p ξ l + p ψ ! . (36)As the final point we have to replace the Poisson brackets with corresponding Diracbrackets. However it is important to stress that the Dirac brackets between the firstclass constraints coincide with corresponding Poisson brackets. Let us demonstratethis claim on the following example { T T ( N ) , T T ( M ) } D = { T T ( N ) , T T ( M ) } −− Z d D z d D z ′ { T T ( N ) , G A ( z ) } △ − ( z , z ′ ) (cid:8) G IIA ( z ′ ) , T T ( M ) (cid:9) ++ Z d D z d D z ′ (cid:8) T T ( N ) , G IIA ( z ) (cid:9) △ − ( z , z ′ ) { G A ( z ′ ) , T T ( M ) } ≈≈ { T T ( N ) , T T ( M ) } (37)due to the fact that { T T ( N ) , G A ( z ) } = − N ( z ) G IIA ( z ) ≈
0. In the same way wecan show that the Dirac brackets { T T ( N ) , T S ( N i ) } D , { T S ( N i ) , T S ( M j ) } D coin-cide with corresponding Poisson brackets. Further, it is also easy to see that theDirac brackets between g ij , π kl coincides with the Poisson brackets again simplyfrom the fact that { g ij , G A } = 0 , { π ij , G A } ≈
0. On the other hand the situation ismore complicated in case of the modes ξ l , ψ l and corresponding conjugate momenta p ξ l , p ψ l . Explicitly { ξ l ( x ) , p ξ k ( y ) } D = { ξ l ( x ) , p ξ k ( y ) } − Z d D z d D z ′ { ξ l ( x ) , G A ( z ) } △ − ( z , z ′ ) (cid:8) G IIA ( z ′ ) , p ξ k ( y ) (cid:9) ++ Z d D z d D z ′ (cid:8) ξ l ( x ) , G IIA ( z ) (cid:9) △ − ( z , z ′ ) { G A ( z ′ ) , p ξ k ( y ) } == δ ( x − y ) δ lk − κ dU dψ l △ − √ gδ ,k δ ( x − y ) , (38)where △ − is defined by the equation Z d D z △ ( x , z ) △ − ( z , y ) = δ ( x − y ) . (39)Then using (34) we find △ − ( x , y ) = 1 √ g ( − κ dU dψ + D − D B ) δ ( x − y ) . (40)9n the same way we find { ξ l ( x ) , p ψ k ( y ) } D = − √ g κ dU dψ l dU dψ k √ g △ − δ ( x − y ) , { ψ l ( x ) , p ξ k ( y ) } D = −√ gδ l, δ k, △ − δ ( x − y ) , { ψ l ( x ) , p ψ k ( y ) } D = δ ( x − y ) δ lk − √ g κ δ l, dU dψ k △ − δ ( x − y ) . (41)Remarkably the presence of the second class constraints implies non-trivial Diracbrackets between metric variables and scalar fields and corresponding conjugatemomenta. For example, { g ij ( x ) , p ξ l ( y ) } = Z d D z d D z ′ (cid:8) g ij ( x ) , G IIA ( z ) (cid:9) △ − ( z , z ′ ) { G A ( z ′ ) , p ξ l ( y ) } == 1 D g ij △ − δ ,l δ ( x − y ) . (42)In the same way we find { g ij ( x ) , p ψ k ( y ) } D = 12 κ D g ij dU dψ k △ − δ ( x − y ) , (cid:8) π ij ( x ) , p ξ l ( y ) (cid:9) D = − D π ij △ − δ ,l δ ( x − y ) , (cid:8) π ij ( x ) , p ψ k ( y ) (cid:9) D = − κ D π ij dU dψ k △ − δ ( x − y ) . (43)Let us outline the results derived in this section. We performed the canonicalformalism for non-local theories of gravity. We found that the Hamiltonian is givenas a linear combination of the first class constraints with standard Poisson brackets.On the other hand the Hamiltonian constraint and symplectic structure defined onthe reduced phase space are very complicated due to the relations (35), (36).In the next section we perform the Hamiltonian analysis of non-local theory ofgravity that is formulated in the Einstein frame. For some purposes it is convenient to transform non-local theory to the Einsteinframe formulation . Recall that under the scaling transformation of metric¯ˆ g µν = Ω ˆ g µν (44) For review, see for example [22, 23]. ( d ) ¯ R = 1Ω (cid:18) ( d ) R − D
1Ω ˆ g µν ∇ µ ∇ ν Ω + D (3 − D ) ∇ µ Ω ∇ ν Ωˆ g µν Ω (cid:19) . (45)Let us consider the most general form of the non-local action (9). Then using (44)with Ω = B − D we can map the action (9) into the form S = Z d D +1 x p − ˆ g (cid:26) κ F ( A, η , η , . . . , η m , ψ , ψ , . . . , ψ n )++ ( d ) R + 1 + D − D B ˆ g µν ∂ µ B∂ ν B − B − D A ++ 1 B ˆ g µν ∂ µ χ ∂ ν A + 1 B ˆ g µν m X l =2 ∂ µ χ l η l − + B D +11 − D m X l =1 χ l η l ++ 1 B ˆ g µν n X l =1 ∂ µ ξ l ∂ ν ψ l + B D +11 − D ξ A + B D − D n X l =2 ξ l ψ l − ) . (46)This is the non-local gravity action formulated in the Einstein frame. Our goal isto perform the Hamiltonian analysis of given action.The simplest possibility corresponds to the situation when ∂ µ A = 0. In this casethe action (46) has the structure S = 12 κ Z d D x √− g [ ( d ) R − ˆ g µν G AB (Φ) ∂ µ Φ A ∂ ν Φ B − V (Φ)] , (47)where G AB is a specific field dependent metric on the field space. This is well knownform of the scalar tensor theory and it is simple task to determine correspondingHamiltonian H = Z d D x ( N H T + N i H i ) , H T = H G.R.T + H scalT , H G.R.T = 4 κ √ g (cid:18) π ij π ij + 11 − D π (cid:19) − κ √ gR , H scalT = 12 (cid:18) κ √ g p A G AB p B + √ gκ G AB g ij ∂ i Φ A ∂ j Φ B + V (Φ) (cid:19) , H i = p A ∂ i Φ A − g ik ∇ j π jk . (48)Then the standard analysis implies that H T and H i are the first class constraintsand their algebra takes the form (22). Recall again that the Hamiltonian vanisheson the constraint surface. 11ore interesting situation occurs in case when ∂ µ A = 0 which corresponds tothe form of the non-local gravity action (23). Following standard analysis we derivethe Hamiltonian in the form H = Z d D x ( N H T + N i H i ) , (49)where H T = 4 κ √ g (cid:18) π ij π ij + 11 − D π (cid:19) − κ √ gR −− √ g κ B D − D F ( A, ψ , ψ , . . . , ψ n ) + √ gB − D A − − D D B √ g p B ++ 1 + D − D √ g B g ij ∂ i B∂ j B − B √ g n X k =1 p ξ k p ψ k −− √ gB g ij n X l =1 ∂ i ξ l ∂ j ψ l − √ gB D +11 − D ξ A − √ gB D − D n X l =2 ξ l ψ l − . (50)As usual we obtain the secondary constraints H T , H i that obey the relations (22).On the other hand the requirement of the preservation of the primary constraint p A ≈ ∂ t p A = { p A , H } ≈ N B D − D √ g (cid:18) κ dFdA − B + ξ (cid:19) ≡ N B D − D G A ≈ , (51)where G A coincides with the constraint (32). Finally we determine the time evolu-tion of the constraint G A ∂ t G A = { G A , H } == BN − κ n X l =1 d FdAdψ l p ξ l − p ψ + 1 − D D ) Bp B ! + 12 κ d FdA v A = 0 . (52)We observe that this equation possesses two possible alternatives exactly as theequation (29).Since the analysis is completely the same as the analysis presentedbelow this equation we will not repeat it.In summary, Einstein or Jordan frame formulation of non-local theory of gravityleads to well defined Hamiltonian systems where the Hamiltonian is given as a linearcombination of the constraints. Due to the fact that these constraints have to vanishweakly it does not matter whether the scalars are ghosts or ordinary scalar fields atleast on the classical level. Acknowledgements:
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