A Toy Model of Gravity

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HeavyParticle
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Joined: Thu Mar 10, 2016 9:25 am

A Toy Model of Gravity

Postby HeavyParticle » Mon Oct 09, 2017 6:22 pm

The following began as a speculative but intuitive investigation into gravity concerning the Penrose Induced Collapse Theory (PICT). Later, I find extensions to the hypothesis which more or less gave me enough material to call it a ''toy model'' of gravity. Since we do not know the exact dynamics of gravity and its role within quantum theory, it is reasonable that this model is very speculative. The investigation has been helped along the way by talented online posters offering advice on the physics, including Matti Pitkanen, a physicist who I consider a friend, who has also offered valuable information throughout.

Nevertheless, even with the existence of such uncertainty, an idea of gravity is on the horizon (at least, I remain hopeful). My personal opinion on this matter was formed very early on from only the first principles of relativity, that is, gravity is not even a force by the true quantum definition - which maybe surprisingly makes gravity on a different league to the rest of the so-called, fundamental forces of nature. Because of this, I became sceptical of gravity being a quantized gauge Boson of spin-2 - after all, gravity is a pseudo force, it does not technically require a graviton mediator.

This is why, in this model, I investigated gravity in the context of the phase space using Von Neumann-like operators. The commutation properties smears the classical vacuum into the quantum - attempts to measure gravity at the atomic and quantum scales are on-going. There have been interesting investigations into trying to attempt to find the influence of gravity on superpositioned rubidium atoms - this experiment shows that gravity (as accelerations in the superpositioned states of two interacting clouds) pulled on the clouds at the same rate as other clouds at different energy levels. As you can see, attempting to measure the actual effects of gravity on the quantum scale, is difficult because it appears to be so weak, at least on the scale we can probe spacetime. I say this, because some theories of gravity hold that it may only become quantum mechanically-significant at Planck scales. Later Wheeler created a concept of this in his quantum vacuum foam hypothesis, which was actually an early theory about the existence of quantum fluctuations.

In this work, I will lay out the foundation to this toy model - we will start off by making clear that this model uses (but does not need to depend) on a possible non-trivial spacetime uncertainty principle that is predicted by both string theory and quantum loop gravity. I link this dynamically to gravity in the following way by using the antisymmetric curvature tensor

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

Where we take \ell to denote a Planck length in the phase space. This has also used the space-time subscripts (x,0). This form above is simply calculated in the normal way from the Christoffel symbols which form two connections of the gravitational field, still following of course, the commutation laws,

[\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]

Which as you can see, has boiled down to three commutation components, which when you write them out, will become the Riemann tensor. In order to give the vacuum an intrinsic property of uncertainty, (something which some literature) has hinted at as requiring if vacuum fluctuations are to exist, we adopt a geometric interpretation of the uncertainty principle from the Cauchy Schwarz L^2 space in which the expectation of the uncertainty is found as the mean deviation of the curvature of the system:

\sqrt{|<\nabla^2_i> <\nabla^2_j>|} \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>

It is also noted here, but will not be worked on, that such a spacetime uncertainty identification as the antisymmetric indices of a curvature tensor will have implications with the Bianchi identities, which will be further true up to three cyclic Christoffel symbols. We will look into a crucial part of this model and that is by merging Anandan's physics with the Penrose model and then finally into my own model - in Anandan's model, he speculated the following energy equation related to the geometry of the system:

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

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

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

To continue forward I first made sense of Anandan's equation - he posits k as some constant of proportionality - in the context at least I have explored, the constant is a correction on the upper limit of gravitational force:

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

It's entirely feasible to replace the Christoffel symbol with a direct replacement of the curvature tensor as we have written it, this reveals a difference of geometries in the following form, using the expectation value of the states and its relation to the PICT will be revealed afterwards,

\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 related to the Penrose model by noticing his model also involves the gravitational binding,

E = \frac{1}{4 \pi G} \int(\nabla \phi - \phi \nabla)\ dx^3

I have shown in the past, you can derive a similar model for the density, where 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 given as

M_{tot} = \int 4 \pi \rho\ dR

And can be interpreted in terms of the energy, which uses the g_{tt} component of the gravitational field

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

And so the difference of the two mass formula, and correcting with a factor of the speed of light squared to make this units of energy, we have

\Delta E = 4 \pi \int \rho c^2(1 - \frac{1}{(1 - \frac{R}{r})})\ dV

Which is just

\rho_{energy} = \rho_{mass}c^2 - \frac{\rho_{mass} c^2}{(1 - \frac{R}{r})}

There is in fact, more than one way to continue with my theory - but I did want specifically the ability to describe entanglement in the phase space - which also means, a concept of information theory. This is my earliest attempt to try and find dynamics for a collapse of a wave function instead of assuming it in inherently an intrinsic process. A completely deterministic theory was important to me, because I found little sense in thinking about the collapse in terms of indeterminism, when conventionally-speaking, the wave functions evolution is completely deterministic.

To move on, I started to investigate different ways to look at the theory - I came across a paper by I. Białynicki-Birula, J. Mycielski, on ''Uncertainty relations for information entropy in wave mechanics'' (Comm. Math. Phys. 1975) contains a derivation of an uncertainty principle based on an information entropy and features here as

- \int |\psi|^2\ln[ |\psi(q)|^2] - \int |\hat{\psi}|^2 \ln[|\hat{\psi}(p)|^2]

which can in fact formally satisfy the difference of the energy expectation of geometry equation we derived not long ago. Notice also, the first term is the entropy related to the state (q) and the difference is taken from the entropy related to the state (p). The investigation then led to the Bure metric, which is

F(\sigma, \rho) = \min_{A^{*}_{i}A_i} \sum_i \sqrt{Tr(\sigma A^{*}_{i}A_i)} \sqrt{Tr(\rho A^{*}_{i}A_i)}

As shown from Venn diagrams, it is possible to show a distinct difference between the classical and quantum entropy by noting that inequalities arise relating that entropies are weaker in the quantum case. The inequality that already exists for the quantum theory is larger by a factor of two,

S(A:B) \leq 2\min[S(A),S(B)]

What is this strange notation? We will come back to this in a moment, because we have discovered something about our theory! I take you back to the equation/inequality we derived near the beginning of this work:

\sqrt{|<\nabla^2_i> <\nabla^2_j>|} \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>

This inequality satisfies the quantum inequality bound

<\psi|\nabla_i, \nabla_j|\psi>\ =\ <\psi|R_{ij}|\psi>\ \leq\ 2\sqrt{|<\nabla^2_i> <\nabla^2_j>|}

This means the gravitational entropy of our system can reach twice the classical upper bound. The factor of two can further be interpreted as qubits. The same kind of situation arises in the Von Neumann entropy, in which the base log of 2 in information theory is almost always calculated using the base 2 log. Now, the strange notation located in the equation a few moments ago, can be understood say, in similar context to the Shannon entropy when written as an equation for information entropy in terms of quantum correlation,

H(A) = H(A|B) + H(A:B)

Here, H(A|B) is the entropy of A (after) having measured the system that become correlated in B. And, further, H(A:B) is the information grained about A by measuring B. As seems well known in literature, these two quantities complement each other so that H(A) is unchanged overall to satisfy the second law.

In a similar context to the investigation into I. Białynicki-Birula, J. Mycielski paper on the ''Uncertainty relations for information entropy in wave mechanics,'' we will now investigate the relative entropy and the probability distributions between them (which can also be seen as the difference taken between two states) and so related to the difference of geometry equations:

D(p|q) = \sum_lp_l \log_2(\frac{p_l}{q_i}) = \sum_l p_l(\log_2 p_l - \log_2 q_l)

The way to view this equation, is as an information gain between two distributions (under observation), so is an expectation difference of two states, which is exactly what the difference in geometry equation is all about.

For the interesting single particle wave function case, you could argue that it is possible in the absence of external forces and other particle dynamics, a single wave function could be capable of collapsing under its own gravitational weight by assuming there is an analogue of the centre of mass for a wave function, which can be interpreted as fluctuating around the absolute square of its wave function. For a single particle, the entropy can be seen as

-\sum^{n}_{i = 1} p_i \log q_i = \sum^{n}_{i = 1} p_i \log n = \log n

which is known as the entropy of q. The inequality exists h(p) \leq h(q) with equality \iff p is uniform (a uniform probability distribution). Obviously, in the gravitationally-induced collapse model, we must rely on a non-equilibrium in the system to cause a mechanical and deterministic collapse.

We also now, expand on the definition of the expectation value in information theory. When you treat geometry like an observable, like we have, the expectation value can be related to the density matrix in the following way:

Tr(R_{ij} \rho) = Tr(R_{ij} \sum_i p_i|\psi_i><\psi_i|) = \sum_i p_i\ Tr(R_{ij}|\psi_i><\psi_i|) = \sum_i p_i\ Tr(<\psi|R_{ij}|\psi>) = \sum_i p_i<\psi_i|R_{ij}|\psi>

This retrives the expectation value of the geometry of the systems for any collection of states \rho. This is a very simple solution and helps the continuation of the toy model for future development. When the systems are correlated, the quantum divergence is

S(A:B) \leq 2\min[S(A),S(B)]

And we have shown that this still remains true for the expectation value for gravity. This has allowed quantum entanglement to enter this theory in a natural way, but since we are closing in on the investigation, it may be worthwhile saying a few things about it. Unlike the classical conditional entropy,

S(a|b) = S(a,b) - S(b)

which remains positive always, while the quantum mechanical equivalent form

S(\rho_A|\rho_B) = S(\rho_{AB}) - S(\rho_B)

is not. The state is entangled if S(\rho_A|\rho_B) < 0 where

S(\rho_A \otimes \rho_B) = S(\rho_A) + S(\rho_B)

and

\rho_{AB} = \sum_i \rho_A \otimes \rho_B

Both these equations are always separable, so if

S(\rho_A|\rho_B) = S(\rho_A)

then it is a separable state no correlations. If instead we have

0 < S(\rho_A|\rho_B < S(\rho_A)

then it is said to have ''classical correlations'' and if

S(\rho_A|\rho_B) < 0

means we have the quantum correlations. Going back now to the energy difference equation I derived from Anandan's hypothesis using the squared Christoffel symbol,

\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

While the wave function may not be time dependent in the Heisenberg picture, we can use a variational principle to vary the wave function - the total variation will split this part up

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

into

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

for both geometries, where the subscript of \delta_{ab} denotes a ''two particle system.''

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

\delta_{ab}< R_{ij}> = \int (<\psi |R_{ij} |\delta_{ab}\psi> + <\delta_{ab} \psi |R_{ij}|\psi> - <\psi |R_{ij} |\delta_{ab}\psi> + <\delta_{ab} \psi| R_{ij}|\psi>)

Even though we think about \delta_{ab} 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.

A little note. It's also possible to normalize the wave function to make it independent of it ~

<\delta_{ab} R_{ij}> = \frac{<\psi| R_{ij}|\psi>}{<\psi|\psi>} + \frac{<\psi |R_{ij}|\psi>}{<\psi|\psi>}

This renormalization however if implemented in my theory, would remove the statistical nature to gravity. That would be quite boring.


The Zeno Effect and a Simple Case of Linear Geometry Operator

The density operator is

\rho = |<\psi|\mathcal{U}|\psi>|^2

which can be rewritten as

\rho =\ <\psi|e^{iHt}|\psi><\psi|e^{-iHt}|\psi>

It is known that for small time intervals the quadratic decay law exists

P \approx 1 - (\Delta H^2 t^2)

Can also be seen as

|a_0|^2 = 1 - (\omega t)^2

where

\Delta <\mathbf{H}> = \sqrt{|H|^2 - <\psi|H|\psi>^2}

which measures the uncertainty in the Hamiltonian density in the state \psi for the system in the phase space. Then we can implement the survival probability in the context of the binding energy equation. With c = 8 \pi G = 1 we have

\Delta \mathbf{H} = \sqrt{|R_{ij}|^2 - <\psi|R_{ij}|\psi>^2} - \sqrt{|R_{ij}|^2 - <\psi|R_{ij}|\psi>^2}

So the geometries themselves depend on the survival probabilities of each system. In this case, we cannot think of the curvature tensor R_{ij} as the anti-symmetric relationship - this will be studied further because we have the ground work. The uncertainties in this last equation can be though of as consisting of two terms:

\sqrt{|\psi|^2 \ln[|\psi(p)|^2] - |\hat{\psi}|^2 \ln[|\hat{\psi}(p)|^2]}

\sqrt{|\psi|^2 \ln[|\psi(q)|^2] - |\hat{\psi}|^2 \ln[|\hat{\psi}(q)|^2]}

So in terms of pure statistics, the Hamiltonian density can be expressible as

\sqrt{|\psi|^2 \ln[|\psi(p)|^2] - |\hat{\psi}|^2 \ln[|\hat{\psi}(p)|^2]} - \sqrt{|\psi|^2 \ln[|\psi(q)|^2] - |\hat{\psi}|^2 \ln[|\hat{\psi}(q)|^2]}

Don't be put off with a wave function existing for the first terms, when there are non featured in |H^2| in this part. This is because it can actually be written as


\Delta <\mathbf{H}> = \sqrt{<\psi|H^2|\psi> - <\psi|H|\psi>^2}


Curvature in Hilbert Space


It became clear to me, it wasn't a simple topic when we just add in curvature to Hilbert space - because of the structure of this space - but I did come across a great paper that will be linked after that helped this post.

Does a Hilbert space have a curvature? No it doesn't - not intrinsically, but it does actually have a geometric structure, simply because it does possess a scalar product. I've shown the Hilbert space in the context of a geometric uncertainty relationship for space, known as the Cauchy Schwarz space. What is important, though not intrinsically possessing curvature, it may indeed have one and I see no reason why it can't.

The linear Hilbert space will only satisfy the natural flat geometry. There is in fact, no easy way to give ''curvature'' to a Hilbert space,

There is a way to describe that through the flat metric

d(\psi, \phi) = |\psi - \phi|

There is a projective Hilbert space \mathbf{C}P^n then we can consider what is known as the distance function

d_{proj}(\psi, \phi) = \min_{\alpha}|\psi - e^{i\alpha}\phi| = \sqrt{2 - 2|<\psi,\phi>|}

This is often used to find the ground state of some Hamiltonian. Interestingly (something I learned) the distance function contains a singularity at a quantum phase transition.

The distance function can be thought of as a metric - this may be a better alternative to the Bure's metric (which I have read some authors) claim that it is not actually a true metric: Neverthless, we have looked at a statistical theory of gravity and (I looked at) Bure's metric has implication for quantum geometry information theory. So there are links here that should not really be ignored without investigation. How true the statement that the Bure metric is not a true metric is, I am unsure. What I can say is that the Bure metric is a ''metric'' which measures the infinitesimal distance between density operators which define the quantum states. The Bure's metric, can also be thought of as a ''statistical distance.'' This has formal similarity to a distance function of the form

d(\psi,\phi) = \arccos(|<\psi|\phi>|)

This measures also, the shortest geodesic between any two states! It is an angle formula, we may consider it also in the form

d(\rho_1,\rho_2) = \arccos \sqrt{F(\rho_1,\rho_2)}

Of course, (iff) the Bure's metric is not a true metric, then neither is this ''shortest geodesic distance'' equation. It will be interesting to read vigorous arguments for and against the Bure's metric as being a real metric or not.

Now... we must go back quickly, and go back to the distance function. Using just single notation now, we have

|\psi - \psi'|

This still doesn't yet describe curvilinear space - this is actually a Euclidean measure of the distance between two states. The length of a curve however can be given as:

\frac{ds}{dt} = \sqrt{<\dot{\psi}|\dot{\psi}>}

This involves the tangent vector \dot{\psi} has lengths that is the velocity which travels in the Hilbert space. I can construct a an equation related the evolution of the Schrodinger equation from this. With c= 8 \pi G = 1

\frac{ds}{dt} = \frac{1}{\hbar} \sqrt{<\psi|R_{ij}^2|\psi>} = \frac{1}{\hbar} \sqrt{<\psi|H^2|\psi>}

and the metric of the solution is

ds = \frac{1}{\hbar}(t_1 - t_2)\sqrt{<\psi|H^2|\psi>}

Anandan, a physicist whome I used an equation that described the difference of geometries (but made it within the context of the curvature tensor) has proposed that the Euclidean length is an intrinsic parameter of the Hilbert space.

Assuming I have done all this right so far, I did find an interesting continuation which uses the Wigner function. The curve of the length can be related to an inequality by pulling out the terms to write a Wigner function. Again, in natural units,

\frac{ds}{dt} \equiv  |W(q,p)|  \sqrt{<\psi|R_{ij}^2|\psi>}\ \geq \frac{1}{\hbar} \sqrt{<\psi|H^2|\psi>}

due to the inequality relationship:

|W(q,p)| \geq \frac{1}{\pi \hbar}

The Wigner function satisfies the following relationships:

\int W(q,p)\ dp =\ <q|\rho|q>

\int W(q,p)\ dq =\ <p| \rho|p>

There are other ways to argue this inequality. For instance, some standard formula

\int <q|\rho|q>\ dq = \int <p|\rho|p>\ dp = Tr(\rho) = \mathbf{1}

and

\int <q|\rho|q> <p|\rho|p>\ dqdp = Tr(\rho)^2 = \mathbf{1}

Because our model deals with the phase space, you can argue there exists a Cauchy Schwarz inequality

2 \pi \hbar W(q,p)^2\ \leq\ <q|\rho|q><p|\rho|p>

or

W(q,p)^2\ \geq\ \frac{1}{2 \hbar} <q|\rho|q><p|\rho|p>

Again, to satisfy our Cauchy Schwarz space. The inequality satisfies the norm through

\int W(q,p)^2\ dqdp \geq \int \int <q|\rho|q><p|\rho|p>\ dqdp = Tr(\rho)^2 = \mathbf{1}




Conclusions

As a preliminary and possibly naive first attempt to view a model for gravity, in the absence of full gauge quantization, personally was interesting. In the phase space, where we use commutation properties to explain the smearing of the classical into the quantum space (appears to be) a lucrative step towards such ideas. The most important thing we found was that the spacetime uncertainty geometry written in terms of the Cauchy Schwarz spacetime yields a relationship identical to the classical upper bound of the deviation of the entropy, in terms of quantum correlation no less. We've also been able to successfully talk about how that expectation enters the theory of information in terms of qubits. The conclusions are simple: It is possible to extend Penrose and Anandan's model to suit the difference of geometries in terms of the curvature tensor, which further gives rise to the antisymmetric properties in its last two indices, which we have treated like quantum operators, which are nothing but the connections of gravitational field. Because the Cauchy Schwarz inequality can be thought of as a geometric interpretation of the uncertainty principle, then we can also conclude spacetime does have a natural mechanism for quantum fluctuations. We also took a look at how the geometries of a system may depend on the survival probabilities, a key component in the idea of the quantum zeno effect. We also show how the latter in this case, applied to the Wigner function and even how it applies to the Cauchy Schwarz space (in theory). A conclusion from the study also shows that curvature is not very well defined in the Hilbert space and so you need to start making tools like the distance function, to explain them.



REF

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

http://www.math.uconn.edu/~kconrad/blur ... pypost.pdf

https://en.wikipedia.org/wiki/Kullback% ... divergence

http://www.tcm.phy.cam.ac.uk/~sea31/tiq ... _notes.pdf

http://www.physik.uni-leipzig.de/~uhlmann/PDF/UC07.pdf

CURVATURE OF HILBERT SPACE AND q-DEFORMED 'QUANTUM MECHANICS'
BINAYAK DUTTA-ROY
Saha Institute of Nuclear Physics, Calcutta 700 064, India




notes

(notes will be added to the main work later)

The expectation can be constructed for the Cauchy Schwarz space in the following form:

<R_{ij}> = \sum_n\ <\psi|a_n><a_n|\psi>a_n

= \sum_n\ <\psi|R_{ij}|a_n><a_n|\psi> = <\psi|R_{ij}(\sum_n\ |a_n><a_n|)a_n|\psi>=<\psi|R_{ij}|\psi>

Which does make use of the completeness relation/theorem. When you square this and solve using eigenstates in this form,

\Delta <R_{ij}> = \sum_n\ <\psi|a_n><a_n|\psi>(a_n - <R_{ij}>)

you will get

\Delta <R^{ij}R_{ij}> = \sum_n\ <\psi|a_n><a_n|\psi>(a^2 - 2a_n<R_{ij}> + <R^{ij}R_{ij}>)

=\sum_n\ <\psi|a_n><a_n|\psi>a^2_n - 2<R_{ij}>\sum_n\ <\psi|a_n>a_n + <R^{ij}R_{ij}>\sum_n| <\psi|a_n><a_n|\psi>

or simply

\Delta <R^{ij}R_{ij}> = <\psi|R^{ij}R_{ij}\psi> - 2<R^{ij}R_{ij}> + <R^{ij}R_{ij}> =\ <R^{ij}R_{ij}> - <R_{ij}R^{ij}>

This is maybe how those eigenstates may come into play but the question is whether it preserves unitarity?

A unitary transformation on |\psi> will preserve the norm, as expected

|\psi'> = \mathcal{U}|\psi>

Unitarity is preserved through the operators and its conjugate

<\psi|\mathcal{U}^{\dagger}\mathcal{U}|\psi> = 1

In our case, we treat the geometry as satisfying completeness (and though Born rule satisfies 1 in normal circumstances, the Born rule is not always unitary), still satisfies unitarity if and only if in our model R^{ij}R_{ij} > 0 and as such, violations are found in R^{ij}R_{ij} < 0.

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

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

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

R_{ij}' = \mathcal{U}^{\dagger}R_{ij}\mathcal{U}

and

<\psi|R_{ij}'|\psi> = 1

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).

Using the Heisenberg picture, with a state which evolves in time, we have the relationship

<R_{ij}(t)> = <\psi(0)|R_{ij}(t)|\psi(0)>

A usual procedure here is to identify the following relationship as well

R_{ij}(t) = \mathcal{U}^{\dagger}R_{ij}(0)\mathcal{U} = e^{(\frac{i}{\hbar}(H t))} \cdot R_{ij}(0) \cdot e^{(-\frac{i}{\hbar}(H t))

The infinitesimal transfromaton for time t \rightarrow dt makes the unitary operator (in standard textbooks) as ~

\mathcal{U}(t) = 1 - \frac{i}{\hbar}[(Ht)]

I assume there are commutation properties in (Ht) \rightarrow [H,t].

On Aruns Doubly Extended Weak Equivalence Principle

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 e^{+iHt}. 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 the 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 production. 

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

m \rightarrow \infty

Then the temperature goes to zero 

T \rightarrow 0

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

K \rightarrow 0

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

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  3 \times 10^{80} 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. Most will have heard of the flatness problem, that is, ''why is the universe so flat?'' If inflation really happened, or some other rapid expansion phase, then this answers it - but with rising opposition to inflation, it appears we may be forced to think outside the box again. But there are arguments that do exist that could question whether space is flat at all.

The density parameter from the Friedmann equations measures the ratio of the observed vacuum density to the critical density. Only when these two quantities are [exactly] the same does the Friedmann equation allow a geometry which would fit Euclidean flat spacetime.

This exact parameter when both terms are equal, would serve what we see in the vast cosmos, since the universe appears to be spatially flat and homogeneous. Or does it? There is an inconsistency which may hint that the large scale homogeneity could be an illusion.

It turns out afterall, that the observed being equal to the critical density doesn't match observation at all. The critical energy (a tool used to explain possible collapse models) is worked out to be five atoms of hydrogen per cubic metre of space. The actual observed density of the matter in the universe, is somewhere between 0.2-0.25 atoms per cubic metre.

Something isn't consistent here. For flat space truly to exist, requires the observed and the critical densities to be exactly equal, but calculation of the actual density of the vacuum is no where near the estimate required to satisfy a flat spacetime model. You could argue the missing matter is made of the non-luminous dark matter entities in the universe, but until a dark matter particle is found, I think this is up for debate.

How is the Spacetime Uncertainty Principle Consistent when Time is not an Observable?

The spacetime uncertainty principle is actually just a reinterpretation of the usual energy-time relationship - it's also not just strictly time as you will see, the dimensions are fixed ct as a spatial component.

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 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.
Last edited by HeavyParticle on Wed Oct 18, 2017 9:56 pm, edited 70 times in total.

HeavyParticle
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Re: A Toy Model of Gravity

Postby HeavyParticle » Mon Oct 09, 2017 6:50 pm

fixed for typos.




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