[Dubbelosix (OP)]

We stated previously that to answer our problems, it may be that the distribution of mass in superpositioning does not give rise to true equilibrium. Penrose's model seems to be based very much on the same situation:

''The superposed lump plays the role of the Schrodinger’s cat. Supposing the gravitational field of the lump and that each lump location separately represents a stationary state and that energy in each case is the same, can we conclude that the quantum superposition of the two lump locations is a stationary state? Penrose using the profound conflict between general covariance principle and superposition principle, asserts that gravity is responsible for an objective reduction of quantum states, accordingly such superposition is not stationary and collapses (Penrose 1996).''

'' General covariance principle implies that in absence of any spatial inhomogeneity in the background potentials, there is nothing in the intrinsic nature of the lump location that allows us to distinguish it from any other lump location, whereas to sense the quantum superposition of lump locations, those locations must be distinguishable. In other words, to have just a single Schrodinger equation governing the evolution of the superposed quantum system, we have to identify those two space–times and according to the general covariance principle there is no canonical way of asserting which point of one space–time is to be regarded as the same point in another. ''


https://link.springer.com/article/10.1007/s40995-016-0024-9

[Dubbelosix (OP)]

Penrose has suggested a graviational self energy related to a collapse time model



And in the Penrose model, the energy is given as



We can derive a more general case that can be used to measure the density variations of spacetime. Deriving the gravitational binding between any coherent gravitational superpositioning state can be given the following way:

The gravitational field inside a radius is given as



and the total mass is



and so can be understood  in terms of energy (where is the time-time component of the metric),



The difference of those two mass formula is known as the gravitational binding energy:



Distribute c^2 and divide off the volume we get:



Were we have used a notation for the energy density. Fundamentally, the equations are the same, just written differently. Notice that from Poisson's formula, in which we notice the same terms entering





So while Penrose suggests calculating the binding energy directly from the gravitational potential there are ways as shown here, to think about it in terms of the gravitational energy density and the gravitational binding between the two.

Now, J. Anandan has postulated an equation with dimensions of energy



He postulates a fluctuation of the connection and proposes the gravitational fields of the superpositioned states may be regarded as peturbations on the background Minkowski spacetime and suggests



Where we have removed Dirac notation because of the sites sensitivity to certain symbols and format. It's an interesting equation that needs to be understood better for it shares formal similarities to the approach we took in a Hilbert space:



As an expectation value on curvature tensor. When we think of a connection like really, what is being suggested, is a covariant derivative and a connection term



and with the indices and extra terms, it just becomes a more complicated object known as the curvature ternsor . In theory, we should be able to replace for the curvature tensor.



http://sci-hub.bz/10.1007/BF02105068

[Dubbelosix (OP)]

''Now, J. Anandan has postulated an equation with dimensions of energy



He postulates a fluctuation of the connection and proposes the gravitational fields of the superpositioned states may be regarded as peturbations on the background Minkowski spacetime and suggests



Where we have removed Dirac notation because of the sites sensitivity to certain symbols and format. It's an interesting equation that needs to be understood better for it shares formal similarities to the approach we took in a Hilbert space:

....''



Let's explore this first equation by Anandan by studying the dimensions. I thought what was supposed to be implied here from his 'proposed' constant of proportionality, was the speed of light to the power of four, which is an equation with dimensions identical to an energy of the form



So while the author did not specify, did they set to natural units and keep the definition of the Newtonian constant in there?

The equation does have non-commutative properties in it should be noted. Taking both equations and looking at the structure





The equations are very similar.

[Dubbelosix (OP)]

The dimensions of his equation



doesn't make much sense to me... I mean, I could very well be wrong or missing something. I did try and construct some kind of meaning out of it though.

So what is the difference between the two equations?





Firstly, I should have noted that the first equation uses a dimensionless interpretation of which you can expand in a series, which can be thought of as ''gravitational corrections.'' Knowing that the operator has dimensions of inverse length squared and since the first term is a gravitational force , we end up with an equation (with dimensions of which is exactly the dimensions of energy. Going to the second equation I am wondering if there is a typo in the equation and really what we want is



can be written also then as



We may identify the squared value of the divergence is the Laplace operator. This means his equation can be written as



Now compare



An integral on the RHS of the first equation will yield



Which does have dimensions of energy iff the connection is of dimensionless form and the missing constant of proportionality is to fix the dimensions. So



Is an energy density and the energy is simply



Is it possible that such a mistake occurred somewhere? Because the other form doesn't make much sense to me.

[Dubbelosix (OP)]

The notation seems a bit odd to me, for if the previous equation is not what he meant then may not be the Laplacian. It seems at least, from his fluctuation equation, it could be like a difference operator, in which case, we don't usually square it. The similarities though, were interesting. Even if it was a difference operator, the dimensions of his equations would not make sense to me.

So... really, we can replace the ... inspiring for the curvature tensor itself. The difference in quantum geometries in a Hilbert space is



*Check this with the early construction!

Where we have used as a difference operator, in the same manner it seems, as Anandan. In his equation, an element volume term is attached to it, why, I am not sure. We have kept it out of this definition. The equation just proposed, is the proving ground of unitarity and whether the system obeys it. This equation doesn't tell you whether it does.

[Dubbelosix (OP)]

If instead now, his equation




Is not treating as the Laplacian, but instead a difference operator, and if now has conventional dimensions of 1/length then an integral yields a true energy equation



Which would make sense of the element volume attached to his curvature/geometry probability difference and in a case where he set . In which case, you should still be able to write the following:



Where we do not square the operator this time.

[Dubbelosix (OP)]

So yeah, the really confusing thing here, and don't worry, I was confused as well, it doesn't actually matter whether the coefficient of the Christoffel symbol is the nabla operator is replaced with a difference operator. The results appear to be the same either way, which is good. We will go back to this description of Anandan's picture later when I have developed more theory.

[Dubbelosix (OP)]

Ah! Just one last thing to wrap up, the equation whcih describes the binding energy of the superpositioned geometry



with as the difference operator.



We have established, what is really meant with equation 1. is



We have argued, that the squared component of the connection can be interpreted in terms of the curvature tensor. This is related to the difference of geometries and that is given now as a full energy equation with

[Dubbelosix (OP)]

You may have noticed, the energy equation that describes the difference in superpositioned geometry ~



Shares the difference between two expectation values of the system:



That coefficient of may indeed attach to that energy, just like a kinetic energy term. So really, when you saw this object: as we have shown, we had already calculated this identity very early on in the work. So the energy equation is compatible in a Cauchy-Schwarz interpretation of spacetime.

[Dubbelosix (OP)]

How to Vary the Expectation

In terms of a Laplacian, it was possible to construct an equation of the form: Again, some of these equations have been pulled out of their code because of site sensitivity to Dirac notation.





which does not have a squared value for the Laplacian . The funny thing is ... again, it all led to the same end anyway. Notice though, even if it was a Laplacian, you would not be squaring the Laplacian because dimensionally, that makes no sense. You could square the nabla operator and argue the wrong notation was used, but since the same notation crops up all over the internet, I must still assume it is a gap in my knowledge.

(\Delta A)^2 = <\psi| A^2| \psi> - 2<A^2> + <A^2> = <A^2> - <A^2>


You can see the equivalent of equation 2. in the quote as


<A> = \sum_n <\psi | a_n > <a_n|\psi> a_n

 = \sum_n <\psi | A | a_n><a_n|\psi > = <\psi| A (\sum_n |a_n ><a_n|) a_n |\psi> = < \psi| A |\psi>


Which makes use of the completeness theorem. To find the alternative version, you square and solve from the form involving eigenstates: Using their notation ~

(\Delta A)^2 = \sum_n <\psi | a_n > <a_n|\psi> ( a_n - <A>)^2

= \sum_n < \psi | a_n><a_n |\psi>(a^2 - 2a_n<A> + <A>^2)

= \sum_n <\psi| a_n> <a_n |\psi> a^2_n - 2<A>\sum_n <\psi |a_n><a_n| \psi> a_n + <A>^2 \sum_n <\psi|a_n> <a_n |\psi>

So that's how those strange extra symbols come into the game, even though they were never implied in the formulation of our Cauchy Schwarz spacetime.

Right... back to the work.

How do you vary the expectation value in the equation



The total variation will split this part up



into



where the subscript of denotes a ''two particle system.''

[Dubbelosix (OP)]

...



where the subscript of denotes a ''two particle system.''

Take careful note though, this last equation is only for one variation in one such term for the expectation - so you will end up with a four component equation in a superpositioned system of two particles or by calculating the binding energy, which also invites the differenece of two expectations.

[Dubbelosix (OP)]

The eigenstate approach gave rise to two extra terms, and we explained why above.

But now the reference, or you won't know what the hell I am talking about.

http://physics.mq.edu.au/~jcresser/Phys301/Chapters/Chapter14.pdf

[Dubbelosix (OP)]

A little note. It's also possible to normalize the variation by the square of the wave function





The reason why we would want to do this is because it would make the expectation independent of the normalization of the wave function. Also expect there to be a variation in curvature tensor on the right hand side. The variation could possible be written in the following way .

[Dubbelosix (OP)]

Right... back to the work.

How do you vary the expectation value in the equation



The total variation will split this part up



into



where the subscript of denotes a ''two particle system.''

Let's construct the full equation now. Expand the equation a bit on the right hand side and setting and also taking as , which seems to be a common practice,



When you vary the wave function for each term, you get the full equation



(again, devoid of Dirac notation due to site sensitivity to latex).

Even though we think about as some differential change between two particle systems, in the context of the right handside, its actually the difference between two geometries of two particles - previous to the collapse state, that one particle is smeared though space. This has been the context of our investigations and how to view gravity.

[Dubbelosix (OP)]

Not sure if all these equations will show, the place seems sensitive to certain latex equations. If it doesn't work, I will take it out of its latex code so you can translate it.


To give a hint in how to do this unification attempt, we have three key equations,

1.



These are the exact Christoffel symbols of the antisymmetric tensor indices .

2.



This was an equation derived by another author, finding the relationship in a different form argued from quantum mechanics. As you will see in key equation 3. the form has similarities to application of a Hilbert space ~

3.

\sqrt{|<\Delta X_A^2>< \Delta X_B^2>|} \geq \frac{1}{2} i(< \psi|X_AX_B|\psi > + <\psi|X_BX_A|\psi>) = \frac{1}{2} <\psi|[X_A,X_B]|\psi>

Again, this is a Hilbert spacetime commutation relationship of operators which has to translate into the gravitational dynamics dictated by key equation 1.

So let's put it altogether, its just like a jigsaw puzzle now. Implemented the Christoffel symbols in approach 1. into approach 2. we get



In the framework of the Hilbert space it becomes - assuming everything has been done correct, takes the appearance of ~


\sqrt{|<\nabla_i^2>< \nabla_j^2>|} \geq = \frac{1}{2} i(< \psi|\nabla_i\nabla_j|\psi > + <\psi|\nabla_j\nabla_i|\psi>) = \frac{1}{2} <\psi|[\nabla_i,\nabla_j]|\psi> = \frac{1}{2} <\psi | R_{ij}| \psi > = \frac{1}{2} < \psi |- [\partial_j, \Gamma_i] + [\partial_i, \Gamma_j] + [\Gamma_i, \Gamma_j]| \psi >


without imaginary number on

I was told by my friend my Hilbert space would be infinitely dimensional, I don't think this is the case at all, after more studying. The L^2 function implies finite results in the Hilbert space and L^2 space has functions which satisfy Cauchy Schwarz inequality. Since we formulated a spacetime theory using the Cauchy Schwarz inequality, it seems reasonable to assume this is finite dimensional.


http://mathworld.wolfram.com/L2-Space.html