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Dubbelosix

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Everything posted by Dubbelosix

  1. It's getting late, made several typo's in the last post. Be back when I can, may not be around tomorrow. anyway, think that's the errors weeded out and fixed. My latex gets sloppy when I get tired. Good night.
  2. We can work it out properly now. The survival probability is [math]\rho = |<\psi|U|\psi>|^2[/math] This is just [math]\rho = <\psi e^{iHt}|\psi><\psi|e^{-iHt}|\psi>[/math] As we have established before. You can expand further for (small) time intervals, it is said to be given as [math]P \approx 1 - (\Delta H^2t^2)[/math] This is the quadratic decay law also written as [math]|a_0|^2 = 1 - \omega^2 t^2[/math] Where [math]\Delta <H> = \sqrt{|H|^2 - <\psi|H|\psi>^2}[/math] That measures the uncertainty of [math]H[/math] in the state [math]\psi[/math] for one of the systems in our phase space. This can be though of in terms of the energy difference of geometries as a Hamiltonian density [math]\Delta H = \frac{c^4}{8 \pi G} (\sqrt{|R_{ij}|^2 - <\psi|R_{ij}|\psi>^2} - \sqrt{|R_{ij}|^2 - <\psi|R_{ij}|\psi>^2})[/math] (So if I have done this right) our system is one that depends on the survival probabilities of each system, while it is a statement itself about the binding energy also. I think this raises interesting situations, if the state of one system depends on the other. The survival probability https://sci-hub.bz/https://doi.org/10.1007/s11005-011-0539-0 (Mordred) I'll certainly take a look soon. Thanks.
  3. 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.
  4. 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 mechanicshttps://arxiv.org/pdf/1210.6348.pdfIn 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.
  5. I did a lot with the equation of state in my own investigations, especially in the non-zero value context which was required for the non-conservation of a dynamically expanding fluid. The zero context of it was the first assumption of Friedmann (maybe influenced by Noether) and it produces the constancy of energy in his theorem. This is why the Friedmann equation is considered by some physicists, as a statement of conservation. Motz has argued, the constancy of energy as spacetime expands is an unfounded assumption. I think the equation of state though is a reasonable description for late cosmology, not too sure when the universe was young and curvature dominated. It is these sistuations that can easily lead to non-conservation, in at least two different ways I know about, both involving on-shell and off-shell matter. Carrol seems to believe the universe does not globally conserve energy but for different reasons, such as no global time means no translation with energy and so no conservation.
  6. 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.
  7. I figured out when I was much younger that the sum of any of the digits of a prime number never come to factors of 9. Some random examples 953 = 9 + 5 + 3 = 17 = 1 + 7 = 8 1087 = 1 + 8 + 7 = 16 = 1 + 6 = 5 3187 = 3 + 1 + 8 + 7 = 19 = 1 + 9 = 10 3691 = 3 + 6 + 9 + 1 = 19 = 1 + 9 = 10 7793 = 7 + 7 + 9 + 3 = 26 = 2 + 6 = 8 And I am quite sure the rule continues. I thought in my younger years it was a discovery, but it turned out there was statement of it somewhere and no doubt has a name which I certainly don't remember now. I think the jury is out, but most mathematicians tend to think it is likely there is some rule that describes them because they do show patterns from time to time, such as this one above - I think there is a rule out there since it is obeying certain principles within its structure. Who knows? Why it avoids factors of nine is interesting, for me at least, because the sum of its components in the multiplicative table show interesting anomalies, such as a palindrome made of prime numbers [18][27][36][45][54][72][81] The difference of 1 with 8 is 7. The difference of 2 and 7 is 5... and so on, produces: 75311357 Which is a palindrome constructed of prime numbers (if you take 1+1) = 2. For some reason, this always interested me, probably for naive reasons. What can be said of it, is that this is the first four prime numbers without running into double digits - again, if and only if you consider the factors of 1 portraying the prime number 2.
  8. Oh I do understand you of course. The many histories is part of applying the wave function to the worldline.
  9. 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.
  10. 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.
  11. 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.
  12. 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.
  13. 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
  14. 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
  15. 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.
  16. 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.
  17. 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.
  18. Done some more, but as always Mordred, will need time to type up later.
  19. Well, initial reports of the Higgs boson that I can remember was that it appears to be nonstandard, that is a Higgs not predicted by the standard model. I haven't read much more than this but did find this article which references that the Higgs has raised questions whether it is standard after all. https://cds.cern.ch/record/1490272/files/plb.726.564.pdf There's loads of things not specifically predicted by the standard model but have been added later. Or there have been extensions or predictions of the standard model, which in their own right, has extended the standard model. http://www.digitaljournal.com/article/328308
  20. I know loads of examples which have been listed in the beyond standard model extensions. Consider the glueball. Again this is classed as an exotic form of matter... The article at wiki on physics beyond the standard model erroneously states that glueballs haven't been discovered, and yet they have. https://www.sciencedaily.com/releases/2015/10/151013103227.htm For me, beyond the standard model physics, was never about breaking physics, or laws, though this could be the extreme case. Here is some very recent news about a possible breakdown in standard model physics: https://press.cern/update/2017/04/lhcb-finds-new-hints-possible-deviations-standard-model What I find interesting, is that the way wiki treats the issue of beyond the standard model seems similar to me. For instance, gravity is listed as beyond the standard model investigations, because it is not accounted for in the standard model, though this isn't meant to mean gravity will use some physics defying principle. Dark matter and Dark energy are both considered beyond the standard model because the standard model, according to wiki, because the standard model does not account for the mass, though you will have posters here argue that all there is, is the standard model - in which case, I ask what is the point of this literature, than to simply notice, the standard model has many new parameters it once never had; extensions to laws that abide physics. Wiki does seem to be clear about listing predictions as part of beyond the standard model physics, this must include discoveries like the pentaquark, gluon and other forms of exotic matter. ''Theoretical predictions not observed[edit] Observation at particle colliders of all of the fundamental particles predicted by the Standard Model has been confirmed. The Higgs boson is predicted by the Standard Model's explanation of the Higgs mechanism, which describes how the weak SU(2) gauge symmetry is broken and how fundamental particles obtain mass; it was the last particle predicted by the Standard Model to be observed. On July 4, 2012, CERN scientists using the Large Hadron Collider announced the discovery of a particle consistent with the Higgs boson, with a mass of about 126 GeV/c2. A Higgs boson was confirmed to exist on March 14, 2013, although efforts to confirm that it has all of the properties predicted by the Standard Model are ongoing.[12] A few hadrons (i.e. composite particles made of quarks) whose existence is predicted by the Standard Model, which can be produced only at very high energies in very low frequencies have not yet been definitively observed, and "glueballs"[13] (i.e. composite particles made of gluons) have also not yet been definitively observed. Some very low frequency particle decays predicted by the Standard Model have also not yet been definitively observed because insufficient data is available to make a statistically significant observation.'' https://en.wikipedia.org/wiki/Physics_beyond_the_Standard_Model
  21. Ok I am sorry, but I didn't say he was retarded, I asked why he was acting like he was when I knew he wasn't. Bad judgement call, sorry.
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