No. A Hamiltonian formulation means a space of states and operators that act on them. In particular a quantized formulation means that it's possible to define a measure on the space of states, that describes geometries that do not solve Einstein's equations of motion-expressed in Hamiltonian form.  That's what quantization means. There's no hint of such properties here. 

A graviton is the 1-particle state of the field defined by the metric, that's all. This field theory is a  gauge theory, with the diffeomorphism group as gauge group. 

While GR is the classical limit of gravity, it's not at all clear of what quantum field theory it is the limit. A major problem is the generic appearance of black hole states, that describe, classically, causally disconnected regions of spacetime. A quantum theory of gravity must be able to resolve such singularities. In particular, point-like objects, of finite mass, would be naked singularities. That's one way of realizing that objects more complicated than points must describe the dynamics of gravity. 

Thank you Stam for taking your time to answer the question. From your statement " A graviton is the 1-particle state of the field defined by the metric, that's all. " I can tell that you are applying the standard model concept of graviton to interpret the Nexus Paradigm graviton . The graviton in the Nexus Paradigm has properties that differ from those of the standard model graviton.  It has also been shown that in the Nexus Paradigm there are no singularities you mention (kindly read the  references provided in the paper ).

Stam Nicolis I have attached a version that includes the Hamiltonian equations and the commutation rules 

Dear Stuart,

"Is this a quantized formulation of General Relativity?"

One should analyze this question from the point of view of physics, not maths.

Physics has experimentally shown that the only (alleged form of) General Relativity that could result in quantum states is given by the following:

The Gravity Probe B experiment tested gyroscopes in satellites, which got an axis orientation change by the gravitational induction of the spinning Earth.

This can only be described by gravitomagnetism, which includes an induction field to gravity, and is entirely analoguous to electromagnetism.

In the annexed paper "Swivelling time...", this has been clarified for the case of spherical/elliptical galaxies with a spinning center (central angular momentum) that are transformed into disc galaxies by gravitational induction.

The fig. 1.2 shows that for orbits that are inclined around 0 rad, pi/4, 3pi/4, or 2pi, the swivelling accelerations towards the equitorial plane are zero. This means that these values will be preferential locations for orbiting stars in disc galaxies.

Hence, there is a macroscopical quantum pattern in gravity here, analoguous to angular quanta in atoms.

From the point of view of gravitons, gravity can be seen as caused by circling gravitons about masses, obeing to the Coriolis (force-)effect. An example is given by the empirical data of the Sun: the rotation speed, mass and radius coincide with this model, as explained in "Is the differential...", and result in the Newtonian gravity.

Moreover, the concept of (circling) gravitons is substantiated by the annexed paper "Relativistic velocity...", where it is shown that masses of the Planck scale can reach the speed of light in gravity fields.

Article Is the Differential Rotation of the Sun Caused by a Coriolis...

Research Swivelling time of spherical galaxies towards disk galaxies

Article Relativistic velocity stabilization of particle sets in gravity fields

Thank you Thierry for your time and suggestion that physics should be interpreted from a physics stand point.  The Nexus graviton induces constant rotational motion on any test particle embedded within it's radius of action just as you point out that aspects of Gravitomagnetism should be found in quantum gravity. Dark matter is a manfestation of this Gravitomagnetic aspect of quantum gravity. 

Any quantum theory of gravity, describes gravitons in the semi-classical limit-that's a standard consistency check of the formalism. In addition to that, it has to describe other spacetime geometries.

Singularities aren't removed-they're resolved-and must reappear in the classical limit.  The proposal presented doesn't seem to do this. 

Okay Stam why are you adamant that singularities aren't removed? GR is an incomplete theory and that was well understood from its inception when singularities arose alongside with  the problem of energy conservation. Again the graviton you are alluding to is the SM graviton and not the Nexus graviton. The Nexus graviton is not a messenger particle but an atom of four space that can expand or contract. It's radius depends on its quantum state and can assumes scales ranging  from the Planck four length to the Hubble four  radius . Gravitational-waves arise in this paradigm in much the same way as sound is generated in matter. 

http://www.worldscientific.com/doi/abs/10.1142/S0218271817500201?journalCode=ijmpd

 Two observations currently in progress are likely to confirm predictions of the Nexus Paradigm. These are the LIGO Collaboration and the Event Horizon Telescope. Nexus predicts that all mergers of BHs will occur at speeds slightly above half light speed with low mass BH mergers occurring at almost exactly c/2. Simulations using GR predict 0.4c and those using Post Parameterized Newtonian (PPN) calculations predict 0.5c. PPN is not appropriate in the strong gravity regime of BHs.

The silhouette of Sgr A* is predicted to be 26 micro arcseconds in the Nexus Paradigm whereas GR predicts 52 micro arcseconds. Previous observations have already shown that the shadow must be less than 37 micro arcseconds.

https://en.m.wikipedia.org/wiki/Sagittarius_A*

The statement that the description proposed has anything to do with quantum gravity is incorrect. The reason is that how a classical gravitational wave induces transitions between quantum states isn't under control. The transition amplitudes aren't ell-defined.

Insofar as the ``Nexus graviton'' is a solution of Einstein's equations, it is a classical background and doesn't imply anything about quantum effects.

Dear Stuart,

Thank you for your appreciation. However, you wrote: "Dark matter is a manfestation of this Gravitomagnetic aspect of quantum gravity."

I am afraid that that is not correct. Dark matter is an invention due to the non-understanding of the velocity curve of disc galaxies by the omission of the effects of gravitational induction, and due to the non understanding of galaxy clusters due to the wrong use of the boundary conditions virial theorem.

I have a simplified paper showing that the velocity curve of a dic galaxy is close to sqrt(GM0/R0) in which the index 0 relates to the data of the central bulge. This is due to the swiveling of all the orbits of a spherical/elliptical galaxy into a disc galaxy, by the gravitational induction of the central angular momentum.

You find the paper annexed. A more detailed paper exists too.

Concerning the galaxy clusters, I explain in the annexed paper why, since Zwicky, the virial theorem has been wrongly used, resulting in a wrong mass definition. Moreover, the gravitational induction can alter the gravity force and the gravitational constant (in the Newtonian notation) will alter as well.

Article Erroneous Use of the Virial Theorem for Elliptical-and Disc ...

Okay thanks Thierry. I have seen your work. Now can you assist in answering the question posed in this thread? 

Stam Nicolis thanks for your opinion, however I disagree with your perspectives. You evidently have not read my work which I have cited in which I address the  problems you cite. Evidently you are a proponent of the standard model graviton and it may be difficult to have a paradigm shift. I am confident that the Event  Horizon Telescope and the LIGO observations will provide evidence in support of the Nexus Paradigm. No quantum theory of gravity thus far other than the Nexus Paradigm  has made testable predictions. 

I agree with Stam Nicolis that you don't really have a quantisation of general relativity, for the reasons he has stated. You seem to write some strange looking Fourier expansions of something... this might be some kind of  'non-commutative space-time' (?).

I am not sure what your \Delta x and \Delta p really are.  What kind of mathematical objects are they?   You define \Delta x as a spinor of some kind, okay it carries a spin rep of the Poincare group, but where is the `quantumness'?  Is the idea that you have an uncertainty principle? Thus, your space-time is divided up into `Planck cells'?  If so, you then want to build a representation of these on some Hilbert space... You would build something like a representation of the `volume' operators or something.  You would start do do `quantum geometry' trying to parallel what you can do in standard quantum mechanics, and then relate this in some classical limit to the Einstein field equations... or there will be little way of knowing you are actually looking at quantum gravity.

One less ambitious idea is to look at deformations of exact solutions to the Einstein Field equations. That is to create some NCG from the classical space-time.  But I don't think this is what you are doing either.

In short,  what is it you have quantised or are trying to quantise?  Some algebra built from Minkowski space-time? I don't quite see where the non-commutativity is.

Your article is not enough for people to really understand what you are trying to do. Well, in truth I can't make a lot of sense of it.

Thanks Andrew you just said that you do not understand my work so kindly read the references provided first to get the best picture. You will find it very interesting. If you cannot access them through the pay wall I will be happy to provide you with preprints.  

@Stuart,

maybe you could answer some of my questions, that way I would be on a better position to read the papers, if your answers spark enough interest. For sure what you have posted here is not enough.

---------------------------------

Okay, I took another look at one of your papers...

So, you realise there are Clifford algebras and in particular we have a Clifford algebra associated with Minkowski space-time.  You then pick a representation, I think the Dirac rep.  I don't understand your expression for

\Delta x^\mu.

The indices don't seem to match. Moreover, it maybe useful to put the spin indices in.  It looks to me that your \Delta x^\mu should have a spin index and that your a^\mu is really a with a spin index.  If so then your \Delta x has a space-time index and a spin index.  This `feels' more like a  Rarita–Schwinger fermion.

I can't get any further than that without you clarifying what sort of object \Delta x really is.

Thanks Andrew . The Rarita -Schwinger equation describes a composite fermion of 3/2 spin. The Nexus graviton is a composite spin 2 boson with each component having spin 1/2. It can be considered as a four vector in Minkowski space or a pulse of four space. The Nexus graviton can be represented as a Fourier transform of the rectangle or pulse function, that is in the form of a wave packet. The wave packet is of course subject to the uncertainty principle . This is to say it is pulse of 4 space associated with a conjugate momentum. It replaces the  point co ordinates in four space that are the source of singularities in GR and QM. The Nexus graviton can assume dimensions ranging from the Planck four length to the Hubble four radius. 

Also take note  that because of the uncertainty principle the four vector traces out a four sphere  (region of uncertainty ). 

You say this, but have not addressed any of my questions....

Okay thanks for showing me the missing spin indice! As you correctly point out \Delta x has a space-time index and a spin index. The spin of each spacetime component is 1/2 giving a total combined spin of 2.

There are of course some conventions on how you place the indices - I would put it lower...

Anyway, I suggest you try re-write the expression for \Delta x. Put in all the indices for clarity and make sure the indices `balance' on both sides of the expression.

I am also not sure about the spin. The object has both space-time and spin indices. This suggests  and more complicated representation like that of the Rarita–Schwinger field.  You have to show carefully how it transforms under the Lorentz group - just do this for the Lorentz algebra to be sure.

Okay thank you for your constructive feedback. I will start working on your recommendations. If you can also find the transformations under the  Lorentz group I will be glad to acknowledge  your  contributions in the final paper. 

No need to acknowledge me... I am just pointing out the obvious things to be sure of before you can really claim anything.

Thinking about the representations, i.e., how things transform under the Lorentz group, you can in a way read that off by the indices - provided the indices are all correct.

In particular, you can consider a rotation by 2pi. For every space-time index the such a rotation goes `all the way round' - I am being slack here, you should be thinking of eigenvalues really.  Each space-time index encodes a spin of one. So, standard vector fields are spin-1, while the metric which has two space-time indices is spin-2. You apply a rotation to `each index' and you `go round twice' as there are two indices. And so on, classically anyway as higher spin fields seem to have problems in  the quantum theory.

By construction, each spin index takes you `round half the way' when you apply a rotation. You can see this with the at first strange looking  factor of 1/2 that appears in the operator S that encodes the Lorentz transformations acting on spinors. So need to apply a full rotation twice to `go all the way round'. This is in part the origin of the idea spin-1/2.

An object with two spin indices on Minkowski space-time would  be spin-1.

You really need to make the distinction between the indices, though on Minkowski space-time the difference is not so big. You can identify the space-time itself with its tangent bundle, and so no real issue. When dealing with curved space-times it is vital to be very clear as we can only identify the tangent space at a point with Minkowski space-time.

Anyway,  your objects \Delta x carry a space-time and a spin index.  This makes me think they are spin -3/2 as this is spin-1 + spin 1/2. Unless , your objects are actually bi-spinors, i.e., they carry two spin indices. But this is also not at all clear from what you have written.

And this is before we think about some Fourier expansion and applying some kind of second quantisation.

I  have been working on it. Results so far show that objects are bispinors 

Λ = e^(ω_{μν}M^{μν }) is the  Lorentz transformation. The vector space is C^4

Yes, the space of Dirac spinors is indeed isomorphic to C^4.  Which makes me wonder why you have chosen that representation? My instinct would be to fix the conventions so that I could choose a real Majoranna representation and keep everything real, but that is just my preference from a geometry perspective.

The inner product \Delta x mu \Delta x mu   is represented as (Ax+By+Cz+Dct)(Ax+By+Cz+Dct) =x2 + y2 +z2 -c2 t2 this generates the Clifford algebra among the coefficients 

I am not sure if what you have written is quite right, but yes,  Clifford algebras come from vector spaces equipped with a quadratic form. In this context, the vector space is Minkowski space-time and the quadratic form is the space-time interval.

But you still have a lot of freedom in picking a concrete representation of this Clifford algebra. This boils down to a choice of gamma matrices.

Basis on experiment with weights weighing the same, when we conected him on electrical weight they masses is smaller from weight taken separately by 5%. Maybe with particles this is the same, and force of gravity is 5% masses of object. And they gravitate by hersel becaouse got some magnetic load, I talk about it with Ouggouti.

Then 2,5% mass of particle and his magnetic load, give us  gravitate. Possible that the nuclear core has four magnetic poles, and electrons and protons gravitate around core. Some electrons and protons going alone and linked to some cores.