# Looking at the Spacetime Uncertainty Relation as an Approach to Unify Gravity

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Abstract

I explore the anticommutating spacetime relation in context of gravity by seeing commutation happen in two connections of the gravitational field, one concerned with space, the other time. We learn nothing revolutionary this time around, but we do explore it in a finite dimensional, Hilbert space-context. I do offer though, in a new context, an equation proposed by Anandan, which can describe the difference of geometries directly related to the $L^2$ space we explore as a Cauchy Schwarz spacetime. My ultimate hope is that it will catch on that the latter gives a natural mechanism for fluctuations in spacetime, as we relate fluctuations to the energy time relationship $\Delta E \Delta t$.

The commutation relationship known as the spacetime uncertainty is established to satisfy a direct interpretation into the antisymmetric tensor,

$R_{\mu,\nu} = [\nabla_x, \nabla_0] = \nabla_x \nabla_0 - \nabla_0 \nabla_x \geq \frac{1}{\ell^2}$

That is, a space $x$ and $0$ (time) notation. I worked out the Christoffel symbols and I calculate in the normal way as two connections:

$[\nabla_i,\nabla_j] = (\partial_i + \Gamma_i)(\partial_j + \Gamma_j) - (\partial_j + \Gamma_j)(\partial_i + \Gamma_i)$

$= (\partial_i \partial_j + \Gamma_i \partial_j + \partial_i \Gamma_j + \Gamma_i \Gamma_j) - (\partial_j \partial_i + \partial_j \Gamma_i + \Gamma_j \partial_i + \Gamma_j \Gamma_i)$

$= -[\partial_j, \Gamma_i] + [\partial_i, \Gamma_j] + [\Gamma_i, \Gamma_j]$

From here, we reinterpreted this in terms of the Cauchy Schwarz inequality to give spacetime an ''instrinsic relationship'' to the uncertainty principle - which may serve as an origin to fluctuations in spacetime - at least this was my motivation - I later discover through more investigation this makes it part of $L^2$ space and thus, a finite dimensional Hilbert space.  The expectation of the uncertainty is the mean deviation of curvature in the system is:

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

$= \frac{1}{2} < \psi |- [\partial_j, \Gamma_i] + [\partial_i, \Gamma_j] + [\Gamma_i, \Gamma_j]| \psi >$

I also speculate in terms of the spacetime uncertainty, anticommutation will exist in the Bianchi identities. The Bianchi identity is true up two three Cyclic Christoffel symbols:

$R_{\sigma \rho [i j]}g^{\sigma \rho} = \partial_i \Gamma_{j} - \partial_j \Gamma_{i} + \Gamma_{i} \Gamma_{j} - \Gamma_{j} \Gamma_{i}$

$R_{\sigma i[j \rho]}g^{\sigma i} = \partial_j \Gamma_{\rho} - \partial_{\rho} \Gamma_{j} + \Gamma_{j} \Gamma_{\rho} - \Gamma_{\rho} \Gamma_{j}$

$R_{\sigma j [\rho i]}g^{\sigma j} = \partial_{\rho} \Gamma_{i} - \partial_i \Gamma_{\rho} + \Gamma_{\rho} \Gamma_{i} - \Gamma_{i} \Gamma_{\rho}$

You can write these three relationships out in the Bianchi identity, we can write the commutation again, on the indices

$R_{\sigma \rho[ i j]} + R_{\sigma i [j \rho]} + R_{\sigma j [\rho i]} = 0$

Again, the last two indices reveal antisymmetric properties. I worked out a static model for superpositionng will not satisfy the fundamental spacetime relationship! Using J. Anandan's equation which I investigated:

$E = \frac{k}{G} \Delta \Gamma^2$

I noted the equation confused me early on, but it seems it is constructed in the following way

$<\Delta \Gamma^2> = \sum <\psi| (\Gamma^{\rho}_{ij} - <\psi |\Gamma^{\rho}_{ij}| \psi>)^2|\psi >$

I think I realized what was implied by Anandans first equation by noticing his missing constant of proportionality is $c^4$. Then an integral of the volume yields the energy

$E = \frac{c^4}{G} \int \Delta \Gamma^2\ dV$

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

$\Delta E = \frac{c^4}{8 \pi G} \int < \Delta R_{ij}>\ dV =\frac{c^4}{8 \pi G} \int <\psi|(R_{ij} - <\psi |R_{ij}| \psi>)|\psi>\ dV$

This is actually related to the difference found in Penrose's model of an induced gravitational collapse in a superpositioned system - albiet, ours is quantum geometry related directly to the Riemann tensor. You may have noticed, the energy equation that describes the difference in superpositioned geometry ~

$\Delta E = \frac{c^4}{8 \pi G} \int < \Delta R_{ij}>\ dV =\frac{c^4}{8 \pi G} \int <\psi|(R_{ij} - <\psi |R_{ij}| \psi>)|\psi>\ dV$

Shares the difference between two expectation values of the system:

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

That coefficient of $\frac{1}{2}$ may indeed attach to that energy, just like a kinetic energy term. So really, when you saw this object: $<\psi |R_{ij}| \psi>$ 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. How do you vary the expectation value in the equation?

$<\Delta E> = \frac{c^4}{8 \pi G} \int <\Delta R_{ij}>\ dV =\frac{c^4}{8 \pi G} \int (<\psi|R_{ij}| - <\psi| R_{ij}| \psi>)|\psi>\ dV$

The total variation will split each two terms,

$<\psi|R_{ij}| - <\psi| R_{ij}| \psi>)|\psi>$

into

$<\delta_{a} R_{ij}> = <\psi| R_{ij}| \delta_{a}\psi> + <\delta_{a} \psi |R_{ij}|\psi>$

$<\delta_{b} R_{ij}> = <\psi| R_{ij}| \delta_{b}\psi> + <\delta_{b} \psi |R_{ij}|\psi>$

where the subscript of $\delta_{ab}$ denotes a ''two particle system.'' So it will become a four-component equation with variations in each term of the wave function. In the main work, we also discussed shortly, my model being related to Penrose's model for the collapse of gravity in a superpositioned state. Penrose has suggested a graviational self energy related to a collapse time model

$T \approx \frac{\hbar}{E}$

And in the Penrose model, the energy is given as

$E = \frac{1}{4 \pi G} \int (\nabla \phi' - \nabla \phi)d^3x$

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 $r = r(0)$ is given as

$\frac{dM}{dR} = 4 \pi \rho R^2$

and the total mass is

$M_{total} = \int 4 \pi\rho R^2 dR$

and so can be understood  in terms of energy (where $g_{tt}$ is the time-time component of the metric),

$\mathbf{M} = 4 \pi \int \frac{\rho R^2}{g_{tt}} dR = 4 \pi \int \frac{ \rho R^2}{(1 - \frac{R}{r})} dR$

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

$\Delta M = 4 \pi \int \rho R^2(1 - \frac{1}{(1 - \frac{R}{r})}) dR$

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

$\bar{\rho} = \rho c^2 - \frac{ \rho c^2}{(1 - \frac{R}{r})}$

Were we have used a notation $\bar{\rho}$ for the energy density. Fundamentally, the equations are the same, just written differently. Notice that $\nabla^2 \phi = 4 \pi G \rho$ from Poisson's formula, in which we notice the same terms entering

$\Delta M = 4 \pi \int \rho R^2(1 - \frac{1}{(1 - \frac{R}{r})}) dR$

$E = \frac{1}{4 \pi G} \int (\nabla \phi' - \nabla \phi)d^3x$

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

Note*

It is also possible to write a version of Anandan's equation like the following

$E = \frac{c^4}{G} \int (\nabla \Gamma)^2\ dV = \frac{c^4}{G} \int \frac{1}{R^2} \frac{d\phi}{dR} (R^2 \frac{d\phi}{dR})\ dV$

This part

$\frac{1}{R^2} \frac{d\phi}{dR} (R^2 \frac{d\phi}{dR})$

Is just another way to write a squared product $\frac{d\phi}{dR} \cdot \frac{d\phi}{dR}$. And of course, this is just $\nabla^2 \phi^2$. We've stated this identity before in an equation - note also, $\phi$ is dimensionless.

Edited by Dubbelosix

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studiot    1155

This is a good one for the long winter evenings ahead.

But please indicate for the sake of those of us who don't do this all the time, the meaning of your bracket symbols.

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5 minutes ago, studiot said:

This is a good one for the long winter evenings ahead.

But please indicate for the sake of those of us who don't do this all the time, the meaning of your bracket symbols.

You mean this?

[a,b]?

This is notation for commutation. It asks whether the process of multiplying in a binary way, two variables is commutative. The notation [a,b] is simply in this context, (ab-ba). Classically speaking, many things never commuted anyway. It wasn't just a special feature of quantum mechanics. When you do speak about commutation and Von Neumann algebra, you end up dealing in what we call the phase space.

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studiot    1155

I hadn't noticed the commutator one so thanks for pointing that out.

I was was thinking more of the Driac bra & ket notation and also the repeated sub/superscript notation, and any othere extreme shorthand notation I may (probably) have missed.

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1 hour ago, studiot said:

I hadn't noticed the commutator one so thanks for pointing that out.

I was was thinking more of the Driac bra & ket notation and also the repeated sub/superscript notation, and any othere extreme shorthand notation I may (probably) have missed.

Without giving you a lecture on the dirac notation, because I am sure you know what that means, in the context we have used it in is

$<\psi|\psi> = \sum_i |c_i|^2 = 1$

where $|c_i|^2= c*_ic_i$ are composed of probability amplitudes, and so the probability to find something in a particular state is given by the mean of the square of the wave function, also known as a quantum collapse. This above, is also known as an expectation value in Hilbert space when acting on an operator, which is the route I investigated.

You ask about repeated indices, the rule is that any repeated indices cancel.

Edited by Dubbelosix

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Mordred    884

So if I'm following correct your in essence breaking the SO(3) group into two unitary Hilbert spaces. You mentioned finite spaces, which is part of the issue with renormalization.

How are you setting the effective cutoffs ? ie the UV cutoff?

The notations above are all in the realm of operators, but part of the issue directly relating is the propogators.

I don't doubt the math above as being correct, LQC does a similar technique in principle but uses a Wicks rotation ( Wilson loop, to quantize units of spacetime)

The only cutoffs I see in the above, is the standard QFT operator IR/UV cutoffs.

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3 minutes ago, Mordred said:

So if I'm following correct your in essence breaking the SO(3) group into two unitary Hilbert spaces. You mentioned finite spaces, which is part of the issue with renormalization.

How are you setting the effective cutoffs ? ie the UV cutoff?

The notations above are all in the realm of operators, but part of the issue directly relating is the propogators.

I don't doubt the math above as being correct, LQC does a similar technique in principle but uses a Wicks rotation ( Wilson loop, to quantize units of spacetime)

The only cutoffs I see in the above, is the standard QFT operator IR/UV cutoffs.

It's a good question. I don't really have an answer (yet).

Let me work on it.

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1 hour ago, Mordred said:

So if I'm following correct your in essence breaking the SO(3) group into two unitary Hilbert spaces. You mentioned finite spaces, which is part of the issue with renormalization.

How are you setting the effective cutoffs ? ie the UV cutoff?

The notations above are all in the realm of operators, but part of the issue directly relating is the propogators.

I don't doubt the math above as being correct, LQC does a similar technique in principle but uses a Wicks rotation ( Wilson loop, to quantize units of spacetime)

The only cutoffs I see in the above, is the standard QFT operator IR/UV cutoffs.

So you have given me a bit of time to think about it, and I have come to see it ultimately depends on the convergence implied from the collapse of the system.  I'm dealing in $L^2$ space and the Hilbert space is $\mathcal{H} = L^2(\mathcal{R})$.

In $L^2$ space, you can have many seqences but converges on

$\int_{-\infty}^{\infty} |a_i|^2 < \infty$

The collapse of the system depends on the collapse time. It has been suggested that Penroses collapse time is too long! A quicker collapse time model is needed, so why not things happening at plank time?

Wheeler suggested quantum foam would occur at a very tiny distance of the Planck scale $10^{-33} cm$, so it wouldn't actually be implausibe to assume any zero point fluctuations must have a cutoff corresponding to the Planck frequency $10^{43} Hz$. Neverthless, the collapse shows you get a finite number, so this is key to the renormalization of this specific theory. That does imply large densities - you might ask why we can't see those large densities and the truth be told, it may all boil down to the lifetime of the fluctuation itself - the short time implies it has only moved a vanishingly small amount and so never becomes anything that is generally visible, at least not with current technology - though, if my spacetime theory of the vacuum is true, that is, that there is an intrinsic spacetime uncertainty that gives rise to particles, it should be possible to create a device to probe and observe spacetime to create those fluctuations. But then you would come to the chicken and egg question - are you creating those fluctuations by looking into the small regions of spacetime, or are they already there?

Edited by Dubbelosix

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Mordred    884

Yes I understand your under L^2 Euclidean norm by the p=2 Cauchy inequality value.

Which if memory serves satisfies the inner space products, have you worked out the Minkowskii inequality?

I'm a bit rusty on Young's

I see from the last established a boundary cutoff except to the operator functions themselves fair enough.

29 minutes ago, Dubbelosix said:

Wheeler suggested quantum foam would occur at a very tiny distance of the Planck scale 1033cm , so it wouldn't actually be implausibe to assume any zero point fluctuations must have a cutoff corresponding to the Planck frequency 1043Hz . Neverthless, the collapse shows you get a finite number, so this is key to the renormalization of this specific theory. That does imply large densities - you might ask why we can't see those large densities and the truth be told, it may all boil down to the lifetime of the fluctuation itself - the short time implies it has only move a vanishingly small amount and so never becomes anything that is generally visible, at least not with current technology - though, if my spacetime theory of the vacuum is true, that is, that there is an intrinsic spacetime uncertainty that gives rise to particles, it should be possible to create a device to probe and observe spacetime to create those fluctuations. But then you would come to the chicken and egg question - are you creating those fluctuations by looking into the small regions of spacetime, or are they already there?

I take it you never looked ar at the Functions for the Feyman S-matrix diagrams? Your verbal descriptive above tells me no.

VP are described under propogators for intent and purpose (internal legs) external legs are operators...

Real particle or states ie observable. minimal 1 quanta of action is required.

So once again where iz your effective Hilbert boundaries to address VP? You have your observable operators but no boundaries to distinquish an excitation from a fluctuation expect via Hamiltonian action under R^n

Edited by Mordred

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Can you expand on what you saying a bit? You mean a Hilbert space in which

$\mathcal{H} = \mathcal{H}^0 + \mathcal{H}^1$

In which you have a 1 particle system in L^2 space, right?

22 minutes ago, Mordred said:

Yes I understand your under L^2 Euclidean norm by the p=2 Cauchy inequality value.

Real particle or states ie observable. minimal 1 quanta of action is required.

So once again where iz your effective Hilbert boundaries to address VP? You have your observable operators but no boundaries to distinquish an excitation from a fluctuation expect via Hamiltonian action under R^n

Right, I am catching on a bit now... there are ways to do this. I remember reading about bounds in the equations. It's all theoretical of course. Even renormalization, though infinities I usually take as a breakdown in the theory.

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Mordred    884

Yes there is ways to handle it but I am trying to distinquish if your handling the problem any different from QFT itself. ie specifically the Dirac relativity equation.

here I'll use this as a time saver.

I didn't find any mistakes in your OP post in the mathematics you posted which is a nice plus. Are you planning on taking this further ? ie stepping it from SO(1.3) to your unitary groups?

Edited by Mordred

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6 hours ago, Mordred said:

I didn't find any mistakes in your OP post in the mathematics you posted which is a nice plus. Are you planning on taking this further ? ie stepping it from SO(1.3) to your unitary groups?

I do feel I have more to do.

I have looked at specific parts of the theory for starters - clearly most of the ground work has been done on this hypothesis. I want to leave some work for other people

I have come across some interesting questions, like early doubts because time cannot be treated like an operator, and of course, the time enters like an operator in this theory - but I have had to be clear, the spacetime uncertainty while though well named because of the structure of the inequality, is also misnomer, to call it a modification of the usual uncertainty principle $\Delta E \Delta t$. This still troubles me.

Time isn't an observable, so hopefully you will understand my discontent. What about coordinate independence? This has also troubled me.

The interpretation of classical to quantum gravity is bothering me - I also want to take this theory into entanglement space. See if the two can be conjoined: I have already sought after a theory with intrinsic spacetime uncertainty.

Just some basic things I have considered.

To ask about the Dirac equation, is a bit more difficult, but not impossible. Just requires more reading and a good brain.  I certainly think its possible to reformulate a Dirac equation in terms of the spacetime uncertainty principle, since the latter here is just a reinterpretation of the usual energy-time relationship for scattered particles.

The interpretation in Planck spacetime is important as well. I wondered whether there are quantum corrections that lead to a quantum redshift

$\Delta L \Delta t g_{00} \geq \frac{G \hbar}{c^4} = \frac{L^2_P}{c}$

This equation involves the dimensionless time-time component of the metric. A metric notation is also $$ds$$ and we will see use of it later.The Planck length is $\sqrt{\frac{G\hbar}{c^3}}$ so we clearly have a squared Plankian structure to the spacetime metric. The non-commutation is expressed on the LHS. A formally similar type of spacetime uncertainty is linked in string theory as well which predicts a spacetime uncertainty linked to the Planck scale.

A simple rearranging of the metric term, and a second order expansion leads to:

$\Delta L \Delta t \geq \frac{(\frac{G \hbar}{c^4})}{[\frac{\phi_1}{c^2} - \frac{\phi_2}{c^2}]} = (\frac{\Delta \phi}{c^2})^{-1}\frac{L^2_P}{c}$

The expansion is analogous to gravitational corrections of the redshift value. In a way, this equation already suggests a link between some spacetime uncertainty relationship to the Planck structure and redshift of the system. The next form of the equation is solved for the uncertainty in a single length, but also makes use of

$\Delta s = \sqrt{g_{00} c^2 \Delta t^2 + g_{xx} \Delta x^2}$

which is a metric term, like we discussed not long ago and is actually just relativity's way to discuss the square of a spacetime interval.

$min[\Delta L] \geq \frac{(\frac{G \hbar}{c^3})}{g_{00}c \Delta t} = \frac{(\frac{G \hbar}{c^3})}{\sqrt{\Delta s^2 - g_{xx}\Delta x^2}}$

Basically we have some Planck structure weighted by a spacetime interval which depends on uncertainties in the metric. The result on the LHS may in a way, be interpreted to mean even intervals on the Planck length can jiggle due to uncertainty.

note

You can expand the gravitational field further

$1 + \phi_1 - \phi_2 - \frac{1}{2}\phi^2_1 - \phi_1 \phi_2 + \frac{3}{2}\phi^2_2 + ....$

For higher corrections to the red shift.

Edited by Dubbelosix

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Mordred    884
11 hours ago, Dubbelosix said:

I do feel I have more to do.

I have looked at specific parts of the theory for starters - clearly most of the ground work has been done on this hypothesis. I want to leave some work for other people

I have come across some interesting questions, like early doubts because time cannot be treated like an operator, and of course, the time enters like an operator in this theory - but I have had to be clear, the spacetime uncertainty while though well named because of the structure of the inequality, is also misnomer, to call it a modification of the usual uncertainty principle ΔEΔt . This still troubles me.

Time isn't an observable, so hopefully you will understand my discontent. What about coordinate independence? This has also troubled me.

The interpretation of classical to quantum gravity is bothering me - I also want to take this theory into entanglement space. See if the two can be conjoined: I have already sought after a theory with intrinsic spacetime uncertainty.

Just some basic things I have considered.

To ask about the Dirac equation, is a bit more difficult, but not impossible. Just requires more reading and a good brain.  I certainly think its possible to reformulate a Dirac equation in terms of the spacetime uncertainty principle, since the latter here is just a reinterpretation of the usual energy-time relationship for scattered particles.

The interpretation in Planck spacetime is important as well. I wondered whether there are quantum corrections that lead to a quantum redshift

Yes I understand your concern with time being treated as an operator, are you aware that position is downgraded to a parameter on the same footing as time under QFT treatments? For other readers (and just in case you aren't) its hard to judge what theories you have studied from a few posts or your understanding in regards to particle physics and its treatments under QFT.

QM we are taught that the symbols $\varphi,\psi$ are wave-functions however in QFT we use these symbols to denote fields. Fields can create and destroy particles. As such we effectively upgrade these fields to the status of operators. Which must satisfy the commutation relations

$[\hat{x}\hat{p}]\rightarrow[\hat{\psi}(x,t),\hat{\pi}(y,t)]=i\hbar\delta(x-y)$

$\hat{\pi}(y,t)$ is another type of field that plays the role of momentum

where x and y are two points in space. The above introduces the notion of causality. If two fields are spatially separated they cannot affect one another.

In QM position $\hat{x}$ is an operator with time as a parameter. However in QFT we demote position to a parameter. Momentum remains an operator.

The reason I asked if you wish to take your model further is that I have a direction for you. Take your model and apply it to the Vacuum solution, then the Newtonian approximation, followed by the Schwartzchild metric.

Why you ask well one aspect of cosmology that can be correlated to an uncertainty of measurement is the cosmological constant.  There is a lot of work and different models looking at this possibility. This may provide some insight into your model at the least.

SO(1.3) is the Lorentz group, hence the above.

I have no idea if your familiar with Clifford and lie algebra ? if not then I can certainly help there as well.

treat your particles as an excitation of a field, apply your creation and annihilation operators under a field treatment,  this will also lead directly into the Klein-Gordon and Dirac equations via the field treatments under QFT ie Langrangian field theory.

Field treatments under symmetry relations where there is no priori of coordinates is extremely useful and grants us a great deal of versatility, after all why should the spherical coordinates, polar coordinates or Euclidean coordinates have a priority.

They are all on equal footing, this is the elegance of field treatments under group theory. (also the basis of GR)

Another member (Strange in another thread) posted this related article to the above, it will provide an example of using the uncertainty principle to apply it to the cosmological constant

as my field is cosmology I thoroughly enjoyed this article and found it extremely useful, you may find it so with regards to your model

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Yes I know Clifford and most useful lie algebra. Of course, only used Clifford in respect to Dirac equation.

As for the interest in the cosmological constant, I too have an interest in it, as I have been studying the Freidmann equation for a while now. I could certainly look at the Dirac equation in terms of the spacetime uncertainty but its not on my first things to do.

I appreciate that you have taken interest in the theory - the problem in taking my model to cosmological interpretations of the cosmological constant is for a full understanding of fluctuations and how they contribute to vacuum energy, if at all. For instance, we may find the fluctuations exist, but exist for such a long time their effects are negligible over the vast distance of spacetime. That would certainly seem the case when we try and measure the vacuum energy - the cosmological constant is something like $10^{122}$ magnitudes too small.

I have came to different conclusions for the vacuum energy problem. I may write up on that here soon.

Edited by Dubbelosix

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Mordred    884
13 minutes ago, Dubbelosix said:

Yes I know Clifford and most useful lie algebra. Of course, only used Clifford in respect to Dirac equation.

As for the interest in the cosmological constant, I too have an interest in it, as I have been studying the Freidmann equation for a while now. I could certainly look at the Dirac equation in terms of the spacetime uncertainty but its not on my first things to do.

I appreciate that you have taken interest in the theory - the problem in taking my model to cosmological interpretations of the cosmological constant is for a full understanding of fluctuations and how they contribute to vacuum energy, if at all. For instance, we may find the fluctuations exist, but exist for such a long time their effects are negligible over the vast distance of spacetime. That would certainly seem the case when we try and measure the vacuum energy - the cosmological constant is something like 10^{122} magnitudes too small.

I have came to different conclusions for the vacuum energy problem. I may write up on that here soon.

That is why I included the particular article I did, Unruh covers that in extensive detail and gives an effective methodology to the IR/UV cutoffs. He goes extensively into the orders of magnitude problem

Edited by Mordred
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I was made aware of some research with some results just out:

''Manipulating rubidium atoms with lasers, scientists led by researchers from Italy gave the atoms an upward kick and observed how gravity tugged them down. To compare the acceleration of normal atoms with those in a superposition, the scientists split the atoms into two clouds, put atoms in one cloud into a superposition, and measured how the clouds interacted. These clouds of atoms behave like waves, interfering similarly to merging water waves. The resulting ripples depend on the gravitational acceleration felt by the atoms.The scientists then compared the result of this test to one where both clouds were in a normal energy state. Gravity, the researchers concluded, pulled on atoms in a superposition at the same rate as the others — at least to the level of sensitivity the scientists were able to probe, within 5 parts in 100 million.''

It seems gravity can affect a wave function!!! This is crucial information, as it tells us gravity does tug on the superpositioned states. This proves that gravity could induce the gravitational collapse.

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A recent article - suggests the same line of investigation I wanted to look at - that is, spscetime can be described fundamentally as threaded together using quantum entanglement. I felt the entangelment of space with space is important in context of an understanding of quantum gravity.

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Mordred    884

Oh my that last link is incredibly misleading when it comes to Ads/Cft lol. Not surprising as one cannot explain degree of freedom reductions too well to the public.

Here you have decent enough math skills, lets save some time

Edited by Mordred
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Posted (edited)

Using the Schmidt decomposition, we have

$|\psi > = \sum_i c_i|\psi_i>_A \otimes |\psi_i>_B$

We can recognize that $\sum_i c^2_i = 1$. You can quantify the amount of entanglement in the system from the entropy:

$\frac{S_A}{S_B} \equiv -\sum_i |c_i|^2 \log |c_i|^2$

This is equivalent to looking at the density matrix in terms of the Von Neumann entropy

$\rho_A \equiv Tr_B |\psi><\psi|$

$\frac{S_A}{S_B} = -Tr_A \rho_A \log \rho_A$

The entropy of entanglement is actually just $E(|\psi><\psi|)$ and it measures the entropy of the pure state $|\psi_i>$.

Because our gravity theory is a theory about gravitional induced collapse by possibly a collection of particles in a superposition, and is not a true static configuration, then there will be a non-zero entropy associated to the system similar to how we viewed this equation:

$\frac{S_A}{S_B} \equiv -\sum_i |c_i|^2 \log |c_i|^2$

This was an important realization because this will tie my entire model together like I wanted - I wanted a theory of spacetime itself, which incorporated the Cauchy Schwarz inequality which is basically a geometric interpretation of the uncertainty principle. I wanted this as a natural mechanism for fluctuations in spacetime, something which is treated in some literature as lacking. My final hope was to incorporate entanglement and thus have some kind of gravity-entanglement as well. Looks like I can do it this way.

Entanglement entropy in QFT would require a definition of the ground state $<0|R_{ij}|0>$ but I much prefer to stay away from this and concentrate on the formalism above which can incorporate a collapse model. That requires perhaps the Shannon entropy $-Tr|\psi|^2\log|\psi|^2$.

Edited by Dubbelosix

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Posted (edited)

I have an equation wrong above,  it should be

$S_A \equiv - \sum_i |c_i|^2 \log |c_i|^2 = S_B$

Whereas, last night, I was writing it out differently as was working from memory, the rest is fine though. I am just working on it right now to try and find a model that makes sense. So this must be corrected before we continue.  Also, you can see the above in another form -

$S_A \equiv - \sum_i |c_i|^2 \log |c_i|^2 = \sum_i |c_i|^2 \log (\frac{1}{ |c_i|^2})$

where once again you can define $|c_i|^2$ as $|\psi|^2$ from the Shannon entropy.

Just a further look into some standard operations.

Joint state is

$\rho_{AB} \rightarrow \sum_i \otimes B_i \rho_{AB} A^{*}_{i} \otimes B^{*}_{i}$

These states are correlated. And it preserves the trac through

$\sum_i A^{*}_{i}A_i \otimes B^{*}_{i}B_{i} = \mathbf{I}$

Disentanglement is found normally through

$\sum_i p_i(\rho^i_A \otimes \rho^i_B)$

Entangelment is defined as

$E(\rho) = min\sum_i p_i S(\rho^i_A)$

Where $S(\rho^i_A)$ can be seen as the Shannon entropy. For a simple case of disentangelment, the shannon entropy will become zero - and for maximally entangled state, it gives $ln2$.

For any pure entangled state with coefficients

$(\alpha |00> + \beta |11>)$

the measure should reduce to the Von Neumann or Shannon Entropy form for

$-|\alpha|^2 ln|\alpha|^2 - |\beta|^2 ln|\beta|^2$

Because my model is about the collapse of a wave function due to gravity and in extended model, about entanglement between two states in some superpositioned geometry, I may find myself coming to find the Bure's metric valuable.

$E(\sigma, \rho) = mim_{A^{*}_{i}A_i} \sum_i \sqrt{Tr(\sigma A^{*}_i A_i)} \sqrt{Tr(\rho\sigma A^{*}_i A_i)}$

This seemed natural since the Bure's metric has implication for quantum geometric information theory.

Edited by Dubbelosix

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Mordred    884
Posted (edited)

Its been awhile since I last looked at Schmidts decomposition the above is in the Von Neumann entropy form correct? The Schmidts equation you gave isn't the standard form and looks like a bipartate mixed state form. Where as Schmidt's has a different form for pure states as well.

Please clarify as it has been some time since I last looked at Schmidt's

(not questioning your usage in the above, just the form posted isn't one I fully recall ).

Have you looked at the pure state for Schmidt's decompositions?

Edited by Mordred

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Posted (edited)
3 hours ago, Mordred said:

Its been awhile since I last looked at Schmidts decomposition the above is in the Von Neumann entropy form correct? The Schmidts equation you gave isn't the standard form and looks like a bipartate mixed state form. Where as Schmidt's has a different form for pure states as well.

Please clarify as it has been some time since I last looked at Schmidt's

(not questioning your usage in the above, just the form posted isn't one I fully recall ).

Have you looked at the pure state for Schmidt's decompositions?

try this link: Tell me what you think

Bure's metric as I have written has a typo, an extra sigma when there should just be a rho  term, which isn't actually density, sigma and rho are the fidelity of quantum states.

Edited by Dubbelosix

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Posted (edited)
On 21/09/2017 at 2:10 PM, Dubbelosix said:

...

which is a metric term, like we discussed not long ago and is actually just relativity's way to discuss the square of a spacetime interval.

min[ΔL](Gℏ/c3)/g00.c.Δt=...

Basically we have some Planck structure weighted by a spacetime interval which depends on uncertainties in the metric. The result on the LHS may in a way, be interpreted to mean even intervals on the Planck length can jiggle due to uncertainty.

...

I'm really not sure of myself, but I think it looks very much like the definition of $l_p ^ 2$

What is different if $\Delta t = t_p$ ?

edit :

oops I don't saw that you have mentionned it

Edited by stephaneww
latex

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Mordred    884

I think your going to have to be incredibly careful here.

The first equation in that link is the standardized Schmidt's decomposition I recall. There is a slew of details you will need to be aware of and account for, to arrive at the equation you posted.

Which does involve Von Neumann entropy, however Shannon entropy also involves Renya entropy.

(getting the idea on why care is needed). We have numerous different methods to define entropy. So greater care is needed in not mixing up these treatments.

35 minutes ago, stephaneww said:

I'm really not sure of myself, but I think it looks very much like the definition of l_p ^ 2

What is different if \Delta t = t_p ?

edit :

oops I don't saw that you have mentionned it

Lol the posts by you two is very similar in your approaches and goals, which has made things interesting for me

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Well, you did say before, if I needed any help with the math, I may take you up on that offer. Any suggestions how to create the Bure metric in terms of the Cauchy Schwarz space inequality? I've been looking into it and while I had confidence (and yes there is literature out there talking about the two subjects as being related) I lost this confidence and realized the terms wa sas you said, something where care was needed.

I am a bit stuck at the moment.

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