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Canonical Analysis of Brans-Dicke Theory Addresses Hamiltonian Inequivalence between Jordan and Einstein Frames

Jordan and Einstein frames are studied under the light of Hamiltonian formalism. Dirac's constraint theory for Hamiltonian systems is applied to Brans-Dicke theory. In both Jordan and Einstein frames, Brans-Dicke theory has four secondary first class constraints, whose constraint algebra is different. The Weyl (conformal) transformation, between the two frames, is not a canonical transformation, addressing quantum mechanical inequivalence as well.


Introduction
It is fairly well known that we never measure in physics absolute quantities, but ratios of absolute quantities. In fact we need to define a unit of measurement u and determine how many times this unit of measurement is contained in the quantity we want to measure. For example suppose we work in natural units where the mass has the dimension of the inverse of length [1]. Be m p the proton mass respect to unit of measurement m u and rescale the unit of measurement by a factor λ −1 , that ism u = λ −1 m u , this implies that in this new unit of measurementm p = λ −1 m p and the ratio [2] stays constant This rescaling appears more intuitive repeating these reasoning on length scales. In fact, in natural units, [1] the above rescaling on the masses implies a length rescaling dx µ = λdx µ and on the metric coefficientsg µν = λ 2 g µν . Therefore [1] invariance of the physical observables under rescaling of units of measurements implies invariance under Weyl rescaling of the metric tensor. This is at the basis of the physical equivalence between Jordan and Einstein frame.
Nowadays the general procedure [2] is to start with a scalar-tensor theory action [3] with the Gibbons-Hawking-York boundary term [4] [5] [6] in what is called Jordan frame where f (φ) is a generic function of φ as well as λ(φ). This theory represents a generic scalar field non-minimally coupled to the gravitational field. Imposing that the action (2) be stationary under the variation δg µν (x) of g µν (x) with the condition δg µν (x) = 0 on the boundary, we get the equations of motion for where .
In strict analogy, equations of motion for φ(x) are derived requiring the action (2) be stationary under the variation δφ(x) imposing δφ(x) = 0 on the boundary In the literature one passes from the Jordan to the Einstein frame [1] [2] through a Weyl transformation of the metric, above mentioned, which now, for convenience, we choose to bẽ g µν being the metric tensor in the Einstein frame. In the Einstein frame the action (2) becomes where varying this equation respect tog µν we get Einstein Equations and varying respect to φ we get the equation for φ(x). As is well known is a solution of the equations in the Jordan frame, then, by construction, (g µν (x, φ), φ(x)) is solution of the equations and in the Einstein frame. Therefore the two frames are physically equivalent provided the scaling relations among observables quantities in the two frames [1] [2]. Anyway this equivalence is still subject of discussion [7] [8] [9]. Recently much work has been devoted to the study of Hamiltonian equivalence between the two frames [10] [11] [12] [13] as well as at quantum equivalence [14] [15] [16] [17] [18]. In the following section we will perform the Dirac's constraint Hamiltonian analysis [19] [20](see also [21] [22] [23] for complementary cases) of Brans-Dicke theory, special case of scalar tensor theory, and we will continue with the same analysis of Branse-Dicke theory in the Einstein frame in order to confront these results in the two frames.

Hamiltonian analysis of Brans-Dicke theory
Brans-Dicke theory [24] is a particular case of (2) when f (φ) = φ and λ(φ) = ω φ [3]: We implement an ADM-decomposition [25], that is we consider a Space-Time (M, g) in which the manifold M is ,topologically, M = R × Σ [20]; R is a one dimensional space, the time direction, Σ is a three dimensional space-like surface embedded in M . The ADM metric tensor g [20] is is the so called lapse function and N i = N i (t, x) are the shift functions. The relative ADM Lagrangian density L ADM [26] is The canonical momenta (π, π i , π ij , π φ ) associated to (N, which show the momenta π and π i associated to the lapse N and shifts N i are primary constraints according to the theory of Dirac's constrained systems [19] [20]. Once we have defined the Legendre transformation (12) to pass from velocities to momenta, we are able to define the Hamiltonian density H ADM knowing the Lagrangian density L ADM This definition holds on the constraint surface defined by the Dirac's primary constraints π ≈ 0 π i ≈ 0 [19] [20] found above (12). Therefore the Hamiltonian density H ADM is and can be written in the following form where the H is the Hamiltonian density constraint, and is just the quantity in square parenthesis of (14) divided by N and H i is the momentum constraint The total Hamiltonian H T [20] is at this point where λ = λ(t, x) and λ i (t, x) are Lagrange multipliers. If we indicate the canonical variables (N, N i , h ij , π, π i , π ij ) generically with (Q i , Π i ) the Poisson Brackets between two arbitrary function A and B of the canonical variables are Following [27], it is possible to show the momentum constraints H i are the generators of the space-diffeomorphisms on the three-dimesional spacelike surface Σ. The constraint algebra among the momentum constraints, and momentum constraints with Hamiltonian constraint can be easily calculated [27] and provides the same results as for Einstein Geometrodynamics [27], [28] As usual, the calculation of the Poisson bracket between Hamiltonian-Hamiltonian constraints is harder . Following [27], the only non-zero terms of the Poisson Brackets originate by non-linear variations of h ij (x) and φ(x) contained in the terms−φ 3 R, ω φ D i φD i φ, 2D i D i φ. Lengthy calculations pursued in analogy to [27] produce where χ i has the following form Notice if φ is constant we get standard Einstein geometrodynamics [28] a [27]. χ i (x) is proportional (12) to the extrinsic curvature K ij , which is not reducible to combination of constraints. We are in the same situation described in [29] {H(x), H(x ′ )} = G(x, x ′ ) for Horava gravity, and in parallel with them we need to preserve the Hamiltonian constraint H on the constraint surface defined both by primary and secondary constraints inhomogeneouṡ equivalent to the following linear inhomogeneous partial differential equation The general solution for this equation equation is obtained by the method of the characteristics [30] [31] [32] log N ( where is a change of coordinates [30] [31] [32]. M is an arbitrary function of ζ 1 , ζ 2 , therefore N is still an arbitrary function and the Hamiltonian constraint H is first class.

Constraint analysis in the Einstein frame and comparison with the Jordan frame
We now consider the ADM decomposition in the Einstein framẽ recalling the Weyl transformation (6) among the metric coefficients in the two frames, we derive the following relations between the lapse and shift functions in the two frames [10] Notice that the lapseÑ and the shiftsÑ i are functions of the field φ(x). The action S of the scalar-tensor action (7) in the Einstein frame for n = 4 for the Brans-Dicke particular case λ(φ) = 2ω φ and f (φ) = φ with the positioñ Applying the ADM decomposition in the Einstein Frame (26), we can derive the ADM Lagrangian densityL ADM , in analogy with the previous paragraph, defining canonical momenta through the Legendre transformatioñ and the canonical Hamiltonian density H ADM is defined in analogy to (13). Performing all the calculations, we get This Hamiltonian H ADM shows the Hamiltonian constraint H is, in parallel to the previous section, the quantity in square parenthesis divided byÑ while the momentum constraints H i are The Hamiltonian H and momentum H i constraints are first class constraints and behave like in standard Einstein geometrodynamics [28] , as it can be easily checked by a straightforward calculation, carefully taking in account thatÑ ,Ñ i and the metricg have now a functional dependence from φ As it is easy to check the secondary first class constraint algebra of the Brans-Dicke theory in the Einstein frame is different respect to the Jordan frame [13]. In particular the Poisson brackets of the Hamiltonian-Hamiltonian constraint in the Jordan Frame (20), differ by the analogous brackets in the Einstein frame (32). This is not, so to say, a surprise because, as is remarked by Hojman, Kuchar, Teitelboim [28] , only non "derivative matter coupling " [33] reproduces Einstein Geometrodynamics, which is not fulfilled by the Branse-Dicke theory in the Jordan Frame. This non equivalence at level of Poisson brackets of the algebra of constraints hints the transformations (27) (29) from Jordan to Einstein frame is not canonical. In fact, if the transformation (27) (29) were canonical then the "symplectic structure"would be preserved, which means that the Poisson brackets among analogous quantities, the Hamiltonian and momentum constraints, would be the same. We recall that in the Hamiltonian theory the transformation (Q i (q, p), P i (q, p)) between two sets of variables (q i , p i ) and (Q i , P i ) is canonical if the "symplectic two form"ω = dq i ∧ dp i is invariant that is ω = dQ i ∧ dP i , which is equivalent to say that the Poisson brackets fulfill the following conditions {Q i (q, p), P j (q, p)} q,p = δ i j (33) {Q i (q, p), Q j (q, p)} q,p = {P i (q, p), P j (q, p)} q,p = 0.
It is quite easy to check that  [28] mainly for the Poisson bracket between the Hamiltonian-Hamiltonian constraint. The same analysis in the Einstein frame shows a Hamiltonian theory equivalent to Einstein geometrodynamics addressing the transformation from Jordan to Einstein frame is not canonical. This difference respect the Lagrangian formalism is due to the fact the Legendre transformation(12) for constrained systems is not a one to one map, having its Hessian determinant equal to zero [35]. This Hamiltonian inequivalence between the Jordan and Einstein frames addresses quantum inequivalence as well. In reference [36], it is shown Hamiltonian quantitation in the minusuperspace case with flat FLRW metric generates two physical inequivalent solutions in the two frames. As regards the path integral [15] [14] [18], it has been already mentioned inequivalence between the two frames. At level of pure speculations, following Dicke's reasoning of the physical equivalence between the two frames, it could be possible, in order to restore full physical equivalence also at Hamiltonian level, to pursue the path of a re-definition of the Poisson brackets like in non-commutative geometry [13].