Hamiltonian Formalism for Nonlocal Gravity Models
Abstract
Nonlocal gravity models are constructed to explain the current acceleration of the universe. These models are inspired by the infrared correction appearing in Einstein Hilbert action. Here we develop the Hamiltonian formalism of a nonlocal model by considering only terms to quadratic order in Ricci tensor and Ricci scalar. We also show how to count degree of freedom using Hamiltonian formalism in this model.
I Introduction
It is certain that universe has entered into an accelerated phase of expansion recently. The simplest theoretical explanation of accelerated expansion is provided by introducing a cosmological constant in the EinsteinHilbert action. However, the addition of such a constant creates a fine tuning problem in the theoryRevModPhys.61.1 . Another way to solve this problem by the presence of nonlocal terms(without introducing cosmological constant) which appear in the Einstein action by taking ultraviolet(UV) and infrared(IR) corrections into account. From cosmological model building point of view nonlocal gravity models have attained various achievements such as (1) it can be employed to study the cosmology in both IR and UV region, (2) it has a well behaved cosmological perturbation theory and (3) the resulting cosmology fits well with most of the cosmological data.
An effective IR corrected Nonlocal gravity model was first developed by WetterichWetterich:1997bz where correction terms like appeared in Einstein Hilbert action. It is found that it doesn’t give rise to correct background cosmological evolution. After a decade Deser and Woodard Deser:2007jk (Nonlocal Cosmology) introduces a non local model of the form along with Einstein Hibert action. One can describe a correct background cosmological evolution with a certain form of Other works in this direction can be found in ref.Barvinsky:2002uf ; Barvinsky:2011hd ; Park:2017zls ; Kumar:2018pkb ; Kumar:2019uwi ; Kumar:2019lzp .
It is well known that higher derivative theories of gravity contain ghost in their spectrum. Ghost appears in their classical and quantum version. This can be deduced from studying graviton propagator in these theories. There are two ways to study the occurrence of ghost in these theories. First way to study the Lagrangian of the system under study and second way to write the Hamiltonian of the system. From the Hamiltonian point of view we look for two points, i.e, first it’s boundedness from below and second is the appearance of Ostrogradsky instabilityLabel1 due to the presence finite number of derivatives higher than two. It is also shown that higher derivative theory of gravity can be renormalizable but only at the cost of unitarityStelle:1976gc ; VanNieuwenhuizen:1973fi . Whereas Ostrogradski instability is not an issue for certain UV corrected nonlocal models containing infinite number of derivatives as it avoids the issue of ghosts and recovers general relativity (GR) at low energies Biswas:2011ar ; Mazumdar:2017kxr ; Biswas:2013kla . As GR is a theory of space time diffeomorphism invariant that the action can contain all possible diffeomorphism invariant term admitting higher derivatives or infinite derivatives of Ricci scalar, Ricci tensor and Weyl tensor.
For a general discussion of the Hamiltonian formalism for the constraint system, refer to Wipf:1993xg ; Dirac:1958sq ; Dirac:1958sc ; Dirac1 ; Matschull:1996up . For UV corrected nonlocal models, such a formalism has been carried out in detail for a nonlocal Lagrangian containing infinite higher derivative terms of Ricci scalar, Ricci tensor and Weyl tensor Biswas:2013cha ; Biswas:2016etb ; Biswas:2016egy ; Biswas:2012bp ; Talaganis:2014ida . In this paper we carry out a Hamiltonian formulation for a toy model with inverse Laplacian operators acting on Ricci scalar and Ricci tensors. We perform a general Hamiltonian formalism in the equivalent scalar tensor form of our theory. We use ADM formalism to separate out spatial and time derivative terms in Lagrangian. Primary and secondary constraints are found and from their Poisson brackets, we obtain the first and second class constraint. This, in turn, helps us to count the number of degree of freedom. Such a formulation for a model only with Ricci scalars has been studied in Kluson:2011tb ..
This paper is organized as the following way, the action and its equivalent scalar tensor action is defined in section 2. In section 3, we review ADM formalismArnowitt:1962hi ; Gourgoulhon:2007ue . Then we convert our scalar tensor action in ADM variable. In section 4, we perform Hamiltonian formalism for a general action where we calculate all the constraints and count the degree of freedom. In section 5, we study the general action for some cases. Finally, we summarise our results in section 6.
Ii Quadratic Nonlocal Gravity
Let us start with constructing a generalized nonlocal action consisting of invariants of Ricci scalars, Ricci tensors and Riemann Tensors. The action can be written asPhysRevD.88.123502 ; PhysRevD.95.043539 ; PhysRevD.98.084040 ,
(1) 
where
(2) 
and is the mass scale associated with nonlocal corrections. We take for our paper. is the inverse of a d’ Alembertian operator. Our notations is similar to that of ref.Teimouri:2018ogt .
Now we divide action (1) in four parts: which are,
(3)  
(4)  
(5)  
(6) 
Further, if we take n=0 then action (1) reduces to quadratic gravity action containing terms upto second order in curvature. For other positive values of nonlocality plays a major role. In next section we write an equivalent action for this nonlocal action.
ii.1 Equivalent Scalar tensor Action
In order to express action (1) in terms of an equivalent action, we wish to translate nonlocal action to scalar tensor one by defining the auxiliary scalar and tensor fields. Now we write the equivalent scalar tensor form of
ii.1.1 Equivalent Action of
ii.1.2 Equivalent Action of
To convert action (4) into its equivalent scalar tensor form first we put , so we get
(8) 
Now we replace R by Q via Lagrange multiplier so we get,
(9) 
By introducing two different set of auxiliary fields and , where such that via Lagrange multiplier and via Lagrange multiplier so on… via Lagrange multiplier . Thus, we can rewrite the action eq.(9), as
(10) 
After rearranging the above action we get,
(11) 
ii.1.3 Equivalant Action of
Similarly for action (5) after substituting we obtain,
(12) 
We replace by by imposing a constraint via a Lagrangian multiplier , so we get an equivalent action,
(13) 
Again we replace by imposing a constraint via Lagrange multiplier , in next step via and so on, finally the equivalent action becomes,
(14) 
Iii ADM Formalism
In order to formulate the Hamiltonian analysis of the above theory first we review basics of 3 + 1 ADM formalism here. Suppose that (M,) is four dimensional manifold, can be foliated by family of space like surface (). In this formalism the 4dimensional metric can be written in terms of induced metric on the 3dimensional surface and normal vector is related to induced metric by
(15) 
where is the time like future directed vector normal to three dimensional spacelike surface, whose norm is,
Any symmetric second rank tensor in 3+1 notation can be decompose as,
(16)  
(17) 
where is purely spatial part, is one normal projection and is two normal projection of tensor .
The line element in ADM formalism is,
(18) 
Here is shift vector and N is the lapse function, defined in terms of metric as,
(19) 
In terms of metric variable, we have
(20) 
In coordinate basis,
Now we can express the projection relations for the curvature tensors in terms of ADM variables. Ricci scalar takes the form
(21) 
Similarly Ricci tensor takes the form
(22)  
(23)  
(24) 
where purely spatial, one normal and two normal projection of Ricci tensor, and is 3dimensional covariant derivative and is Extrinsic curvature tensor and it’s Lie derivative.
iii.1 ADM Decomposition of Action
In this section we decompose actions (7), (11) and (14) into ADM variables. We have already shown decomposed form of the curvature tensor in terms of ADM variables in previous section.
iii.1.1 ADM Decomposition of and
We combine and as both contain Ricci scalar term in a single action
(25) 
next we analyse the decomposition of each term on and .

Decomposition of :
contain second order derivative of metric. However this second derivative of metric can be eliminated by performing integration by parts, so using eq.(21),(26) After simplification,
(27) 
Decomposition of becomes:
(28) 
After some mathematical calculation, we get,
(29) 
By using eq.(27) and eq.(29), we obtain the final form of action (25),
(30) 
iii.1.2 ADM Decomposition of
Similarly here while writing the actions eq.(14) in 3+1 formalism, we need to decompose all the symmetric second rank tensors() into components that are tangent and normal to the hypersurfaces using projection relations is provided in eq.(17).

Decomposition of :
(31) where
is purely spatial part and and are one normal, two normal part of respectively.

Decomposition of :
(32) where,
, and are spatial, one normal and two normal decomposition of respectively.

Decomposition of :
(33) 
Decomposition of is,
(34) where,

Decomposition of :
Here note that contain second order derivative of metric. However this second derivative of metric can be eliminated by performing integration by parts. The part of the action we are interested in here, is(35) The 3+1 decomposition of is given by,
(36) where
Then by using eq.(24), takes the form
(37) After eliminating the higher derivative term we can write the action () as, (see appendix A for details)
(38) 
Decomposition of :
In this case by integrating by parts, action reduces to(39) The full analysis of decomposition of term is given in the appendix B. The final decomposed expression of is,
(40)
A similar expression can be obtained for Hence the action becomes,
(41) 
The complete decomposed form of action becomes,
(42) 
Note that the above action is a function of ) and their time and space derivatives.
Iv Hamitonian Analysis
iv.1 Hamitonian for
Having expressed in terms of and their space and time derivatives, we are now ready to carry out the Hamiltonian analysis of this action. Here we introduce canonical conjugate momenta corresponding to each variable. Canonical momenta with respect to set are
Here we used the sign for showing the primary constraint and they valid only on constraint surface . Hence the primary constraints are summarised as,
It can be observed that the Lagrangian density does not contain the time derivative of and . Hence they are no longer dynamical variables. Hamiltonian density is given by,
(43) 
By splitting along and direction, we obtain,
(44) 
with
(45) 
and
(46) 
Now the Hamiltonian density is given by,
(47) 
where are Lagrange multipliers. and total Hamiltonian becomes,
(48) 
iv.1.1 Classification of Constraints
Here we classify the constraints according to their nature. Our primary constraints are .
Next the secondary constraints are determined by: , , On the constraint space , the time evolutions of and vanishes weakly as
(49) 
and
(50) 
These condition turn fix the Lagrange multipliers , and there will be no tertiary constraints. Now we have identified the primary and secondary constraints. Then we categorise them into first and secondclass constraints. We evaluate Poisson bracket between primary constraint with total Hamiltonian.
(51) 
Then time evolution of become,
(52)  
From the equation of motion for , we derive
(53) 
for Therefore we conclude that the are Lagrange multipliers and further we obtain other primary constraints which are
(54)  
(55) 
Moreover and are primary constraints since the Poisson brackets of these quantities with total Hamiltonian vanish weakly by using their equation of motion. Whereas is a second class constraint since,
(56) 
All other Poisson brackets are obtained as,
(57) 
In summary, we have four first class constraints and two second class constraints and
iv.1.2 Degrees of Freedom
We use the general formula to count the number of the physical degrees of freedom,
(58) 
Where,
A = number of phase space variables,
B = number of secondclass constraints,
C = number of firstclass constraints.
For our nonlocal model, we have
(59) 
where L runs from 1 to . This amounts to
(60)  
This is expected as we have infinite number of derivative terms in the action.
iv.2 Hamiltonian Analysis of
In this section, we focus on the Hamiltonian analysis for the action in eq.(42). First we write canonical momenta of each of variable appearing in the action as