Some new apsects of quantum gravity

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with additional symmetry breaking terms produced by the presence of higher order cumulants of the current.This kind of approximate symmetry breaking is known to give masses to massless particles or more generally, corrections to the masses of already massive particles and we illustrate this idea in the context of interactions of the gravitational field with a random electromagnetic field being regarded as the current.This interaction is the standard Maxwell action used in general relativity.The drawback of our approach to quantum gravity is that is its not diffeomorphic invariant since we have chosen our frame to be always synchronous.Further work on how one can incorporate interactions of the gravitational field with a random non-Abelian gauge field is in progress which becomes important because it generates non only quadratic but also cubic and fourth degree terms in the gauge field when it interacts with gravity.
1 Quantum gravity in the synchronous frame, some perturbative calculations for the equal time commutators The independent components of the metric are just six in number: φ = (φ r ) 6 r=1 = {g rs : 1 ≤ r ≤ s ≤ 3} since our coordinates are synchronous, ie, chosen so that the four conditions g 00 = 1, g 0r = 0, r = 1, 2, 3 are satisfied.The Lagrangian density for φ, namely the Einstein-Hilbert Lagrangian has the form L(φ, φ ,0 , φ ,r ) = (1/2)F 1,rs (φ)φ r,0 φ s,0 +(1/2)F 2rksm (φ)φ r,k φ s,m +F 3rsm (φ)φ r,0 φ s,m The position fields are φ = (φ r ) 6 r=1 .The momentum field conjugate to the position field φ r is P r = ∂L/∂φ r,0 = F 1rs (φ)φ s,0 + F 3rsm (φ)φ s,m let ((G rs (φ))) = ((F 1rs (φ))) −1 Then, we find that the velocity fields are given by φ r,0 = G rs (φ)P s − G rk (φ)F 3ksm (φ)φ s,m or in matrix notation, where ∇ = (∂ r ) 3 r=1 is the spatial gradient operator.In order to make φ ,0 selfadjoint after quantization, we replace the above by φ r,0 = (G rs (φ)P s + P s G rs (φ))/2 − G rk (φ)F 3ksm (φ)φ s,m The equal time CCR's in matrix notation are [φ(t, r), P (t, r ) T ] = iδ 3 (r − r ) so it follows that [φ(t, r), φ ,0 (t, r ) T ] = iG(φ(t, r))δ 3 (r − r ) The field equations where N (φ) is of the form We can expand the nonlinear functional φ → N (φ) as Volterra series: where δ is a small perturbation parameter.We also expand the field solution as where φ 0 , the zeroth order field satisfies gives us After applying the boundary conditions on u(r) the possible "eigenfrequencies" ω assume only discrete values and we can express the solution as Note that if u n (r) is an "eigen-solution" corresponding to the frequency eigenvalue ω(n), then ūn (r) is an eigensolution corresponding to the frequency eigenvalue −ω(n).Thus, taking into account the fact that φ 0 (x) is self-adjoint (which corresponds in classical field theory to a real field), we can better express the above expansion as The zeroth order term in the commutation relation Remark: We are assuming that Planck's constant h is very small and actually appears on the rhs of the above commutation relation.Hence, G(φ 0 ) must actually be replaced by G(< φ 0 (t, r) >) where it then follows that the zeroth order commutation relation is where now G 0 (t, r) is a c-number field.Let us consider the particular case when < φ 0 (t, r) > is independent of time.Then we have [φ 0 (t, r), φ 0,0 (t, r ) T ] = iG 0 (r)δ 3 (r − r ) It follows then that writing that the above commutation relations are satisfied iff Now observe that the u n s satisfy We are assuming that the ω(n) s are real.Taking the conjugate of this equation gives Multiply the first equation by ūm (r), integrate over space, then multiply the second equation by u n (r), integrate over space, subtract the second from the first, integrate by parts and assume that the u n s vanish on the boundary.We get Therefore since now c(r, s)∂ r ∂ s is a self adjoint operator and −ω(n) 2 are its eigenvalues with corresponding normalized eigenfunctions u n (r), we have the result from the spectral theorem that u n s form a complete orthonormal basis for the spatial domain within which the field is enclosed.Then, and n u n (r)u n (r ) * = I 6 .δ 3 (r − r ) These equations imply We therefore modify the CCR to and derive In view of the orthonormality of the u n s, this is equivalent to requiring that This is equivalent to requiring that 2 Quantum gravity in N dimensional space-time,

Hamiltonian formulation
The metric tensor is g µν (x) where 0 ≤ µ, ν ≤ N − 1. x 0 is time and x r , r = 1, 2..., N − 1 are the spatial coordinates.There are N coordinate conditions and these coordinate system can be chosen so that g 00 = 1 and Having done so, the metric tensor now has just N 2 independent component which we denote by φ r (x), r = 1, 2, ..., N (N − 1)/2.This is called a synchronous system of coordinates and the proper time element in these coordinates is given by dτ 2 = dt 2 − g rs (x)dx r dx s We write The Einstein-Hilbert Lagrangian density then has the form r=1 is the spatial gradient operator.The canonical momentum vector (φ is the canonical position field) is given by and hence the Hamiltonian density is The Euler-Lagrange field equations are This equation expands to give (ie we, separate out the components in this field equation that are linear in the space-time partial derivatives and those that are nonlinear in the same) where We shall be assuming that the term N 1 (φ) that is quadratic in the field spacetime partial derivatives are small.We shall in addition, be assuming that where C is assumed to be large while N 2 (φ) is assumed to be small.Without loss of generality, C(r, s) T = C(s, r).We shall also assume that where B is a large constant matrix decomposed as ((B(r))) N −1 r=1 and N 3 (φ) is small.Then, the above field equations can be expressed as where Now we come to the discussion of the CCR.Since for x 0 = y 0 , we have It is clear that by assuming C and B to be respectively the constant parts of , it follows that N (φ) has an expansion in φ that begins with quadratic terms in φ, ie, we can write where K n has an n th order Volterra expansion: 3 How do you take into account quantum noise while formulating a quantum theory of gravity ?
hint: Consider the metric field φ(x) ∈ R 6 in a synchronous frame with Lagrangian density that is a quadratic form in φ ,0 , ∇φ with coefficients that can be complicated nonlinear functions of φ.Express it as Taking the adjoint of this equation, it immediately follows that F (r, r ) = F (r , r) ie F is a Hermitian kernel and hence admits the spectral expansion where λ(k) ∈ R and φ k s form an orthonormal basis for L 2 (R 3 ).Now let A k (t), k ≥ 1 be an infinite sequence of annihilation processes so that they satisfy the CCR and hence also the quantum Ito formula Then, we can write provided that we assume λ(k) ≥ 0∀k.Now we compute the noise modified Lagrangian of the gravitational field as where the unperturbed Lagrangian is Assuming that the amplitude of quantum noise is small, it follows that upto quadratic orders in the noise amplitude, assuming that this Lagrangian density as above is of quadratic orders in the spatial and temporal derivatives of the field φ,the Hamiltonian density can be computed using the Legendre transform as follows: so that upto linear terms in the noise, Retaining upto linear terms in the noise, this Hamiltonian density approximates to the form where the last term H 1 is linear in W, W ,0 , ∇W and linear quadratic in φ, ∇ ⊗ φ, P .It follows that the Schrodinger equation taking into account the linear components in the noise will have the form after spatial discretization, where B(t) is a quantum annihilation process in the language of Hudson and Parthasarathy.It should be noted that W ,0 is white noise and hence W ,0 dt = dB(t) after spatial discretization.This means that the Hudson-Parthasarathy qsde now contains apart from quantum Brownian differentials in addition terms proportional to the Brownian motion processes themselves which reflect the presence of the terms W and ∇ ⊗ W . Here, φ is the position vector and P rhe corresponding momentum vector arising from spatial discretization of the fields.S(φ, P ) is the quantum Ito correction term and is given by and is required to ensure unitarity of the evolution.Here, is the gravitational Hamiltonian in the absence of noise and by calculating φ, P from the linearized solution to the Einstein field equations, these can be expressed as polynomials in the graviton creation and annihilation operators.Remark: We can also take into account quadratic terms in the noise to give a more accurate description of the unitary evolution.Quadratic terms in the noise can be expressed in terms of the conservation process of the Hudson-Parthasarathy quantum stochastic calculus.If we take Fermionic quantum noise also into account arising from contributions from the Dirac action, then we obtain a supersymmetric signal and noise theory for the joint evolution of system and bath.
4 Stochastic problems in quantum general relativity based on the quantum effective action, symmetry breaking caused by higher order cumulants of the random current If J(x) is a random current field with mean M 1 (x) = EJ(x) and higher moments The quantum effective action should then be based on the functional Note that we can write are the central moments of the random field J.The computation of the quantum effective action for a field φ having action S[φ] should be based on Z(M 1 , C r , r ≥ 2) by fixing C r , r ≥ 2 and taking the Legendre transform of Z w.r.t M 1 .Equivalently, in terms of the cumulants of J, and we could write are the cumulants of the random field J and is the cumulant generating functional of J.Note that D 1 (x) = M 1 (x) is the mean of J(x).We could now calculate the quantum effective action for the field φ for fixed values of D r , r ≥ 2 by applying the Legendre transform w.r.t D 1 alone.The quantum effective action is defined by where The extremum above is attained when It is clear that the optimal value of D 1 , namely D 10 is expressible as a function of φ 0 , D r , r ≥ 2. We now derive as usual the quantum equations of motion for the quantum effective action and prove a result that defines the amount of gauge symmetry that is broken when the classical action without the current has a gauge symmetry, in terms of the cumulants D r , r ≥ 2. This model then gives us a method to introduce approximate symmetry breaking due to the presence of randomness in the current field and hence to calculate the masses acquired by particles that represent the field φ in terms of the cumulants D r , r ≥ 2. We first derive the quantum equations of motion: in view of the defining relation for D 1 that extremises iW + D 1 .φ0 as in (1).Suppose now that the classical action S[φ] as well as the path measure Dφ are invariant under the infinitesimal gauge transformation Then we get by replacing the path integration variable φ by φ that Since → 0, this gives us If J were a non-random field, then this equation could be expressed as which in view of the quantum equations of motion derived above for the special case of nonrandom J, would give where J 0 is that current for which < φ > J ) = φ 0 .This means that the quantum effective action is invariant under the infinitesimal gauge transformation Only when ∆(φ) is a linear functional of φ does it follow from this equation that the quantum effective action is invariant under the same gauge transformation ∆(φ 0 ) for which the classical action is invariant since then In the random case, even the nonlinear gauge symmetry of the quantum effective action is broken.To estimate by how much this is broken, we write where where we have defined Then, by a change of path integration variable and using invariance of the classical action and the path measure under the gauge transformation .∆(φ)(x),we get Dividing by and taking the limit → 0, we get where for any functional f [φ] of the field φ, we have defined On the other hand, we observe that by the quantum equations of motion derived above in the random current field case, where D 1 has been computed in terms of φ 0 , D as above, ie, in such a way that the classical-quantum average of φ equals φ 0 .Equivalently, we can write The lhs of this equation gives us the change in the quantum effective action under the quantum gauge transformation φ 0 → φ 0 + .< ∆(φ)(x) > D1 and therefore the rhs gives us the amount by which randomness in the current field causes the gauge invariance of the quantum effective action to be broken.The rhs can thus be used as a correction to the quantum effective action that leads to massless particles acquiring masses or massive particles to have their masses changed.
[2] Consider the Einstein-Hilbert action for the gravitational field.As seen earlier, it has the form where +φ T ,0 C r (φ)φ ,r summation being over the spatial indices r, s = 1, 2, 3. Here, we are assuming a synchronous frame so that g 00 = 1, g 0r = 0 which implies that the metric has just six independent components which we denote by φ.The interaction of the metric field with a random electromagnetic field can be represented by the interaction Lagrangian where D(φ(x)) is a function of only the metric field φ(x) and not its partial derivatives while F = ((F µν )) is the random electromagnetic field.The total Lagrangian of the gravitational field φ interacting with a fixed external random electromagnetic field F is thus Our aim is to calculate the quantum effective action for the gravitational field by assuming that the electromagnetic field F as mean zero and hence In view of this problem, it is instructive to deal with the problem of defining the quantum effective action of a field when it interacts with a classical random current field with all the cumulants of the current field being specified.
Spontaneous symmetry breaking and approximate symmetry breaking in quantum gravity.
5 Propagator computation of a nonlinear field theory using differential equations for time ordered vacuum expectations The equations of motion for the gravitational field in a synchronous frame are expressible in the form Define the gravitational propagator Then, we get Now we can express φ k,00 (x) appearing within the time ordered vacuum expected value on the rhs in terms of φ m,r0 (x) and φ m (x), φ m,r (x), φ m,rs (x), φ m,0 (x).
The troublesome factor here is φ m,r0 (x) because it involves a time derivative which causes it not to commute with the other factors.However, this factor occurs linearly in the field equations and hence its contribution can be evaluated easily as follows: The other troublesome factor here is the one involving φ m,0 (x) which appears nonlinearly in the term F k (φ m (x), φ m,0 (x), φ m,r (x)).Taking all this into consideration, we obtain the following pde for the graviton propagator as We can now make the following approximations: [a] Replace φ(x) by < φ(x) > in C 1 , C 2 , (b) Consider only terms linear in φ n , φ n,0 , φ n,r in F k .This approximation corresponds to neglecting higher that quadratic products in the propagator.However, to determine corrections to graviton mass, we must compute cubic and higher order correction terms also in the propagator.These contributions are in practice evaluated using perturbation theory to express the solution for φ(x) to the field equations in terms of the linearized solution which is expressed in terms of Bosonic creation and annihilation operators satisfying the CCR.However in the quadratic approximation to the propagagtor, we have This approximate graviton propagator equation can be used to evaluate approximately the contribution of nonlinear self interaction of the graviton field to the generation of graviton mass.