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

[Dubbelosix]

A summary elsewhere, and duly noted Mordred, certainly for future discussions. 

 

http://www.physicsgre.com/viewtopic.php?f=10&t=127412

[Dubbelosix]

Done some more, but as always Mordred, will need time to type up later.

[Dubbelosix]

Going back to my Hilbert space investigations, it was possible to construct the expectation in the following form in a Cauchy Schwarz space,

[math]<A> = \sum_n <\psi | a_n > <a_n|\psi> a_n[/math]

[math] = \sum_n <\psi | A | a_n><a_n|\psi > = <\psi| A (\sum_n |a_n ><a_n|) a_n |\psi> = < \psi| A |\psi>[/math]

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

[math]\Delta <A^2> = \sum_n <\psi | a_n > <a_n|\psi> ( a_n - <A>)^2[/math]

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

[math] = \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>[/math]

or simply

[math]\Delta <A^2> = <\psi| A^2| \psi> - 2<A^2> + <A^2> = <A^2> - <A^2>[/math]

This is how the eigenstates come into the game, even though they were never implied in the formulation of our Cauchy Schwarz spacetime this though, is a standard way of calculating them. The same eigenstates could be theoretically implied in our own investigations into gravity.

In a Hilert space, Choosing an orthonormal basis of each such subspace, 
in which they are mutually orthogonal eigenvectors with distinct eigenvalues, it is possible to choose an orthonormal set of eigenvectors which 
most of you will recognize as 

[math](\psi_1>, |\psi_2>, |\psi_3> ... |\psi_n>)[/math]

These set of eigenvectors spans the Hilbert space, which actually has a meaning; It means the orthonormal set of eigenvectors is complete. 

If the expansion coefficient is [math]c_n = <\psi_n|\psi>[/math], the unit operator appears in the standard equation

[math]|\psi> = \sum^{\infty}_{n=1} <\psi_n|\psi>|\psi_n>[/math]

in which the unit operator is

[math]\mathbf{I} = \sum^{\infty}_{n=1}|\psi_n><\psi_n|[/math]

The unit operator is related to the Unitary operator which continues the completness in a unitary way. We can also talk about the spectral theorem for the model, in which in this case [math]Q[/math] is an observable, then

[math]Q = \sum_n q_n|q_n><q_n|[/math]

The eigenvectors of an observable constitte the basis states for the phase space. In hindsight, it has been noted in the work before that we are treating curvature as an observable. 

I'll get into the unitary operator bit later.

Edited October 12, 2017 by Dubbelosix

[Dubbelosix]

As I promised, a little on Unitarity is preserved through the operators and its conjugate

[math]<\psi|\mathcal{U}^{\dagger}\mathcal{U}|\psi> = 1[/math]

In the case of a non-linear operator like ours, still satisfies unitarity if and only if in our model [math]R^{ij}R_{ij} > 0[/math] and as such, violations are found in [math]R^{ij}R_{ij} < 0[/math].

Indeed, if the state satisfies all the above, including unitary, then the expectation should satisfy the norm

[math]<\psi|\mathcal{U}^{\dagger}\ R_{ij}\ \mathcal{U}|\psi> = <\psi'|R_{ij}|\psi> = <\psi'|\psi'> = 1[/math]

(I think I have this right, correct me if I am wrong). This means we can define

[math]R_{ij}' = \mathcal{U}^{\dagger}\ R_{ij}\ \mathcal{U}[/math]

and

[math]<\psi|R_{ij}'|\psi> = 1[/math]

If you define the unitary operator as a time operator (a non-trivial one unfortuantely), then you can choose either the Heisenberg or Schrodinger picture (whether the functions or the observable depends on time).
 

Some credit to mordred though, because something they said reminded me of different things that allowed me to link it this way.

 

Edited October 12, 2017 by Dubbelosix

[Mordred]

 Excellent, will double check em but don't see any errors. Here is the thing once you established your operators you have effectively quantized a vector and correspondantly your vector field treatments. You have established the above to a coordinate basis, deviations from the coordinate basis has transforms we can employ.

You have closed  (including boundary confinement) your Hilbert space, checked orthogonality via Cauchy, thus establishing the Kronecker delta affine connections.   The groups these metrics follow all follow the right hand rule via Hilbert, This must be preserved throughout your metrics.  Also understanding how the projection operators is defined is always a plus plus. Which you have defined in the above...

 I would study Euler Langrene, and Hamiltons under the above before worrying about 4d. Lets make sure you understand how the above works under path integrals.  In particular at the quanta level, in terms of coupling to range of force relations. I have an older thread on this to save time. 

 

http://www.scienceforums.net/topic/106004-useful-fundamental-formulas-of-qft/

 

Though I haven't gotten to the path integrals yet (too many projects :P).

(Once again +1 for your diligence in seeking the proper understanding. No amount of heuristic treatments/ descriptives will get you there. The only path that will is under math). You will also find it won't matter what physics topic your reading. Your understanding improves exponentially to all physics by understanding the mathematics.) 

 Mathematics is universal even in theories. 

I might need to check you hermitean conjugates, but will have to wait till I can focus on it.

edit: no its fine, notation caught me for a second.

With regards to last article and thread, a unitary group that preserves the length also preserves the particle changes with regards to conservation.

(recommended study, skew Hermitean) to above

Edited October 13, 2017 by Mordred

[Dubbelosix]

I wonder if Hestenes geometric algebra could somehow  be implemented, I will look over this in the next few days. I studied his theory a while back but can remember most bits of it so I will have a head start.

[Mordred]

Not sure, don't recall ever running into Hestenes. However I cannot see why not.

You now know any orthogonal group can be made unitary, which is hermitean. Hermitean being symmetric. You also now know how to define length preservation. So any vector is also hermitean and length bound via above.

Also you know how the projector operators are derived via the above. (Though their is several other operators that do not follow the above ie trace operator). Its a short hop to unitary operators, but the trace operator also involves the above.

So I don't see why Hestenes cannot be treated under Hilbert spaces.

(That's the advantage of learning the above mathematics, you can now look at applying it to any physics theory). Lol gives you a major advantage over a vast majority of posters, laypersons etc,

 

The above applies to any QM topic naturally but can also be applied to classical, though for relativity the renormalization issue. ( you can better understand that via the above) Think curve form fitting using discrete units vs curve fitting without discrete units. ( ie divergences of path integrals)

Edited October 14, 2017 by Mordred

[Mordred]

There is a better way to look at this

[math]<\psi|\mathcal{U}^{\dagger}\ R_{ij}\ \mathcal{U}|\psi> = <\psi'|R_{ij}|\psi> = <\psi'|\psi'> = 1[/math]

try the Heisenburg picture using the [latex]\mathcal{H}[/latex] for the Hamilton.

[latex]\langle A\rangle_\tau=\langle\psi(0)|e^{+\mathcal{H}\tau/\hbar} Ae^{-\mathcal{H}\tau/\hbar}|\psi(0)\rangle[/latex]

where [latex]\tau[/latex] denotes proper time.

use the Langrene density and Hamiltons.  You will have a far easier time understanding the path integrals.

https://en.wikipedia.org/wiki/Heisenberg_picture

Edited October 14, 2017 by Mordred

[Dubbelosix]

Yes, I actually have this last identity written up here: Again, (everything) else is noted.

 

http://www.physicsgre.com/viewtopic.php?f=10&t=127412&p=198855#p198855

[Dubbelosix]

Semi-classical gravity does infer a situation which complicates the subject of unitarity within black holes physics. The evolution of two states after forming the black holes are identical, leading to a mixed state obtained through integrating the thermal Hawking radiation states. It leads to the information paradox.

The problem with this is that the final states are identical - we cannot recover the initial state of the evolution just by knowing the final state, even in principle. This contradicts unitarity evolution in quantum mechanics

https://arxiv.org/pdf/1210.6348.pdf

In principle unitarity preserves the ability to recover the initial state if we know the final state by applying a subject we have talked about, the inverse of the time evolution [math]e^{+iHt}[/math]. I can't stress enough, how important it would be to find a clear picture for this within the context of black holes, since, the consensus now is that information is not lost inside of black hole (though I argue cosmologically) information does not need to be preserved as an exact quantity and curvature dominance in early cosmology leads to interesting non-conserved cases, such as irreversible particle product. 

I will certainly contemplate the issue. There was work by Arun who showed you can extend the equivalence principle for cosmological consequences - I extended it further to show it was consistent for observers inside the universe. The principles are simple and intuitive and maybe surprising:

Sivaram and Arum have noted that those relationships are further enhanced by the Von-Klitzig constant and/or the Josephson constant which are used in superconductor physics - black holes are indicated to be diamagnetic, excluding flux just like a superconductor. Truth be told we do not know what the inside of a black is like, we know it stores its temperature on the horizon, presumably with the rest of the black holes information. 

 

Aruns extended weak equivalence is an argument which goes like this: To make the temperature of a black hole go down, you need to add matter to the system. Using the following approximation we have

 

[math]m \rightarrow \infty[/math]

 

Then the temperature goes to zero 

 

[math]T \rightarrow 0[/math]

 

And for a black hole with infinite mass, the curvature tends to zero as well!

 

[math]K \rightarrow 0[/math]

 

As I have stated before though, you cannot really have a system like a vacuum reach absolute zero, when the vacuum is not perfectly Newtonian. To add to his extended weak equivalence, assume the following ~

The radius of a black hole is found directly proportional to its mas  [math](R \approx M)[/math]. The density of a black hole is given by its mass divided by its volume [math](\rho = \frac{M}{V})[/math] and since the volume is proportional to the radius of the black hole to the power of three [math](V \approx R^3)[/math] then the density of a black hole is inversely proportional to its mass radius by the second power [math](\rho \approx M^2)[/math].

 

What does all this mean? It means that if a black hole has a large enough mass then it does not appear to be very dense, which is more or less the description of our own vacuum: it has a lot of matter, around  [math]3 \times 10^{80}[/math] atoms in spacetime alone - this is certainly not an infinite amount of matter, but it is arguably a lot yet, our universe does not appear very dense at all. So this shows Aruns principle is consistent with the very structure of spacetime itself in terms of density and the observers that measure it from inside of it.

 

Though it may not be entirely obvious to posters why cosmological principles like these could have consequence for our understanding of quantum gravity - but really, understanding black holes and their relationship with nature, could turn out to be the key to understanding key principles about quantum gravity itself!

If the principle is taken seriously, and these limits purport to non-physical situations (as infinities should be treated in my opinion) then these limits are telling us only part of the larger picture. What I believe, from my core understanding of physics, is that the temperature of a universe can never reach zero and so can never satisfy a situation where a universe gets large enough that there are no thermal degree's of freedom left and as a result, has a vanishing curvature (tends to flat space). If the Friedmann equation is taken seriously, then observed density does not match predicted density and so does not satisfy flat space without adding in new parameters or making some new assumptions about the fundamental nature of the vacuum itself. 

Here's a paper which has alluded to quantum geometry supepositioning and talks about the issue of quantum gravity and Hilbert space. 

 

https://arxiv.org/pdf/1704.00066.pdf

Edited October 15, 2017 by Dubbelosix

[Dubbelosix]

I also want to consider a new definition for the opening equation:

 

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

 

Which was my application of classical geometry to quantum phase space. A quantum operator for the area [math]\mathbf{A}[/math] can be constructed to express its spectrum. For any arbitrary sets of half integers, a final condition can find a relationship to the Planck phase space with eigenvalues of 

[math]R_{\mu \nu} = [\nabla_x\nabla_0 - \nabla_0 \nabla_x] \geq \frac{1}{\ell^2} \sum_{i}\ (\sqrt{n_i(n_i + 1)})^{-1}[/math]

 

For insight into this approach, the reference is found here: http://cgpg.gravity.psu.edu/people/Ashtekar/articles/qgfinal.pdf

 

It keeps open other possibilities to investigate the Eigenvalues of the geometric Planck phase space.

Edited October 15, 2017 by Dubbelosix

[Dubbelosix]

Mordred.... Does quantum tunnelling destroy a world line?

[Mordred]

hrmm that is an excellent question,

I would have to say yes in similar manner that an acceleration causes new world lines.

[Dubbelosix]

I assume it doesn't, because the only case of worldlines being destroyed is in black hole physics.

Oh ok... you think differently? I'll need to look deeper then.

[Mordred]

Under rotation via rapidity ( ie accleration) the particle no longer follows the same worldline.  A worldline is under constant velocity, a change in velocity via acceleration causes new worldlines so I would think quantum tunneling would cause a similar affect as it will cause a change in velocity ie no longer constant under freefall.

5 minutes ago, Dubbelosix said:

I assume it doesn't, because the only case of worldlines being destroyed is in black hole physics.

Oh ok... you think differently? I'll need to look deeper then.

sounds like this is describing all possible worldlines to a specified observer ie at rest and maximum redshift.

[Dubbelosix]

Isn't this key to the actual mechanism or (understanding) maybe of tunnelling? Maybe just mine...

 

... I have a simplistic view of it - its a situation where, in classical cases, a particle may not have enough energy to overcome a barrier. Then in quantum mechanics, this isn't always the case, owed probably to the uncertainty principle. I have a personal opinion then that will cloud my judgement, because I don't really think anything is random and that maybe linked to the idea that tunnelling may not actually break a world line, but as I said, a very simplistic view and I think I am wrong.

[Mordred]

Particles can and do change worldlines all the time. Remember GR is a freefall geometry freefall is a constant velocity. So anytime the velocity of a particle alters regardless of cause. It will follow a new worldline. Ie New worldlines must be calculated via rapidity boosts and rotations.

A barrier tunnelling would certainly qualify

Edited October 15, 2017 by Mordred

[Dubbelosix]

Am I wrong to think that the destroying of a worldine is not equal to information loss? This is key to where I am heading.

If the worldlines change, like you say, this seems different to the total destruction of a world line leading to those information paradoxes inside black holes. 

[Mordred]

Don't think of it as a single worldline but all possible worldlines between event and observer. ie killing vectors of a metric. This is where the debate drops in on your different coordinate systems involved in information loss. Some killing vectors are artifacts of the metric, ie a horizon is an apparent horizon. 

The majority of this article discusses the killing vectors and touches on Hawking radiation etc 

http://arxiv.org/abs/1104.5499 :''Black hole Accretion Disk'' -Handy article on accretion disk measurements provides a technical compilation of measurements involving the disk itself.

http://arxiv.org/abs/1104.5499 :

Edited October 15, 2017 by Mordred

[Dubbelosix]

Oh I do understand you of course. The many histories is part of applying the wave function to the worldline. 

[Mordred]

exactly, another way to think of it is spatial infinity via equation 2 of this article.

http://export.arxiv.org/pdf/1704.06223

 

though hadn't read the article for quality etc, the formula is accurate.

[Dubbelosix]

Thank you for the link.

[Dubbelosix]

Some interesting information: It is believed that 

'' the fourth dimension is manifested in observable three dimensions as the curvature path of a moving infinitesimal (test) particle''

 

https://en.wikipedia.org/wiki/Five-dimensional_space

 

In the space and time uncertainty, though it is interpreted normally in terms of the results from scattered particles, it may also turn out that time functions like an observable with true units of [math]ct[/math] because of this very simple understanding of how the fourth dimension of space is considered an observable as the curvature experienced in three dimensional space.

[Dubbelosix]

As stated before, I was looking into an extended equivalence principle and was investigating the destruction of the worldlines inside of black holes. Here is a paper that illustrates finding methods of unitarity preservation for entangled particles for black hole physics.

 

https://link.springer.com/article/10.1007/s10701-016-0014-y

 

''Here, we argue differently. It was discovered that spherical partial waves of in-going and out-going matter can be described by unitary evolution operators independently, which allows for studies of space-time properties that were not possible before. Unitarity dictates space-time, as seen by a distant observer, to be topologically non-trivial. Consequently, Hawking particles are only locally thermal, but globally not: we explain why Hawking particles emerging from one hemisphere of a black hole must be 100 % entangled with the Hawking particles emerging from the other hemisphere. This produces exclusively pure quantum states evolving in a unitary manner, and removes the interior region for the outside observer, while it still completely agrees locally with the laws of general relativity.  ''

 

I was looking for explanations outside of those that lead to information paradoxes. It seems they argue they have a way to preserve unitarity inside of the black hole. Recall what I said on the issue of unitarity and black holes;

 

 

''Semi-classical gravity does infer a situation which complicates the subject of unitarity within black holes physics. The evolution of two states after forming the black holes are identical, leading to a mixed state obtained through integrating the thermal Hawking radiation states. It leads to the information paradox.

The problem with this is that the final states are identical - we cannot recover the initial state of the evolution just by knowing the final state, even in principle. This contradicts unitarity evolution in quantum mechanics

https://arxiv.org/pdf/1210.6348.pdf

In principle unitarity preserves the ability to recover the initial state if we know the final state by applying a subject we have talked about, the inverse of the time evolution '''

Paper by t'Hooft no less.

Edited October 16, 2017 by Dubbelosix

[Dubbelosix]

I know a lot of this seems like scribbling... but that's probably because it is. Just as always, looking for simple ways to continue with the toy modeI - I have even went as far to consider a geometric zeno effect! It should be no surprise that the entanglement process (something we have searched for a gravitational interpretation even in this toy model) could be related to the zeno effect (a concept of measurements over time). If our assumptions of gravity hold so far, we can build some picture of it. The density operator is as always:

[math]\rho = \sum_n P_n|\psi><\psi|[/math]

[math]|\psi><\psi| = \mathbf{I}[/math]

and trace operations are just

[math]Tr(R_{ij}) = \sum_n <n|R_{ij}|n>[/math]

The expectation value of the measurement can be calculated  from the case for pure states

[math]<R_{ij}> = Tr(R_{ij}\rho) = Tr(\sum_n P_n|\psi_n><\psi_n|R_{ij}) = \sum_n P_n\ <\psi|R_{ij}|\psi>[/math]

Where the trace of the density operator satisfies

[math]Tr(\rho) = 1[/math]

with a geometric spectral resolution

[math]R_{ij} = \sum_n a_n |a_n><a_n| = \sum_n a_n P_n[/math]

where [math]|a_n>[/math] is an eigenket and [math]a_n[/math] the eigenvalue and:

[math]P_n = |a_n><a_n|[/math]

which is just using the density operator form since [math]Tr(\rho) = 1[/math] satisfies the same completeness or normalization condition as [math]|\psi><\psi|[/math].

Let's just say a little something about the physics of mixed and pure states. A pure state is simply a quantum system is denoted with a vector [math]|\psi>[/math] in the Hilbert space. A statistical mixture of states is a statistical ensemble of independent systems. 

The survival probability (the same probability you ascribe to atoms in a quantum zeno effect set up) 

[math]P_{+}(t) = <\psi_{+}|R_{ij}(t)|\psi_{+}>[/math]

(note, this is just one system).

In which the probability depends on the time and the number of measurements [math]N[/math] which is given by

[math]P_{+}(t,N) = |\alpha_{+}(0)|^2\ e^{-\Lambda t}[/math]

where [math]\Lambda[/math] is the decay rate and the [math]\alpha[/math] is a notation in the two particle system which we will show below. There are ways of course to affect the probability of decay so that they can be completely suppressed. The reason why systems like an atom ripe to radiate away its energy can be affected in such a way, is because the measurement process disturbs the atom in such a way that it rearranges the electrons back into its most stable orbits. The rate in which you make the measurements is crucial - such as, if you leave the measurement after the half life of the atom, the system will be likely to experience an anti-zeno effect. Another way to view it, is that it affects the time evolution of the system. Note now, that an initial system tends to be described by quantum superposition

[math]|\psi(0)> = \alpha_{-}|\psi_{-}> + \alpha_{+}|\psi_{+}>[/math]

In Fotini Markopoulou's toy model of intractions in a Bose-Hubbard space, a state can be a superposition of interactions. For example, consider two systems in the state:

[math]|\psi_{AB}> = \frac{|10>\otimes|1>_{AB} + |10>\otimes|0>_{AB}}{\sqrt{2}}[/math]

This state describes the system in which there is a particle in [math]A[/math] but no particle in [math]B[/math], but also there is a superposition between [math]A[/math] and [math]B[/math] interacting or not. This next state:

[math]|\psi_{AB}> = \frac{|00>\otimes|1>_{AB} + |11>\otimes|0>_{AB}}{\sqrt{2}}[/math]

represents a different superposition, in which the particle degree's of freedom are entangled with the ''graph.'' In other words, Fotini's model shows you can accomodate the entangelment of matter even to geometry! We may never come to use her model, but it is interesting because if the physics (is at least correct in principle) then we can come to expect similar cases within our own model - though we must keep in mind, the Bose-Hubbard model itself is about the interaction of spinless bosons and our model is just a simple look into the Hilbert space.

http://pages.uoregon.edu/svanenk/solutions/Mixed_states.pdf

https://en.wikipedia.org/wiki/Bose–Hubbard_model

https://sci-hub.bz/https://doi.org/10.1016/S0375-9601(01)00639-9

https://arxiv.org/abs/0911.5075

https://arxiv.org/pdf/0710.3914.pdf

 

If the survival probability is constructed for the difference of two systems 

[math]P_{+,-}(t,N) = (<\psi_{+}|R_{ij}(t)|\psi_{+}> - <\psi_{-}|R_{ij}(t)|\psi_{-}>) e^{\lambda t}[/math]

...gives us a form of the Anandan difference of quantum geometries in terms of the survival probabilities. Remember, in Anandan's model, he speculated the following energy equation related to the geometry of the system:

[math]E = \frac{k}{G} \Delta \Gamma^2[/math]

It has also been shown in literature that the difference of those geometries can be written like

[math]\Delta <\Gamma^2> = \sum <\psi|(\Gamma^{\rho}_{ij} - <\psi| \Gamma^{\rho}_{ij}|\psi>)^2|\psi>[/math]

I made sense of that equation in the form:

[math]\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[/math]

 

So of course, all these relationships to the density operator and the expectation and all related subjects will be important in the future. The survival probability of the geometry is something I'd like to work on as a new idea. 

Edited October 17, 2017 by Dubbelosix