Dubbelosix

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  1. I realised while attempting to sleep there is an important interpretation in the work and once again, its because you mentioned something relevant. I remembered that the property [math]<\psi|U A U|\psi> = <\psi'|A|\psi> = <\psi'|\psi'> = 1[/math] Holds if [math]A[/math] is a linear operator. So, while in my past work I considered the geometry as the observable, it would mean we'd have to consider a linear geometry in this case. Which related back to your mentioning, it may be best to look at the non-linear cases now for the more juicy structures of the theory. I had a look at some non-linear wave equations before already... but will take a much greater look into it and see if I can see anything.
  2. 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.
  3. 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.
  4. 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.
  5. 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.
  6. The end of the quantum vacuum catastrophe ?

    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.
  7. 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.
  8. The Cause of Prime Numbers ( Suggestion)

    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.
  9. Oh I do understand you of course. The many histories is part of applying the wave function to the worldline.
  10. 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.
  11. 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.
  12. 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.
  13. Mordred.... Does quantum tunnelling destroy a world line?