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  • Post #43

Dear Tobias,

compliments for your very interesting paper! I'm especially interested in the simulation of Dirac equation by graphene, since this is connected to my work. Can you really consider the carbon atom as a gate in a quantum-computational simulation of Dirac in 2+1 dimensions? In such case I'm indeed very curious about the unitary transformation of the gate!

In the meanwhile I discovered that I didn't see your reply to my answer in my thread, and I answered to it. It seems that you are right. In order to recover an isotropic velocity of light in the analog coordinate system, one needs another way, maybe the thickness of events?

Cheers

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    • Post #44

    Dear Tobias,

    Wisdom is more important than imagination is more important than knowledge for all the we know is just an imagination chosen wisely.

    Please read Theory of everything at your convenience posted by me in this contest.

    Who am I? I am virtual reality, I is absolute truth.

    Love,

    Sridattadev.

    Dear Tobias

    Your essay is very interesting, I think the idea of emergence have not been study very seriously until now but models like the one you expose or others based on a discrete computational basis, show that this is the central point. On my essay, I try to explain this emergence from a different perspective but I think there is a very closed connection with your ideas. I will like to hear what do you think about it.

    Regards,

    J .Benavides

    Dear Mauro,

    excellent question! I think something like this is indeed possible. In a second quantized formalism, the electrons in the tight-binding approximation to graphene would be modeled as follows: take one qubit for each lattice site, i.e. at each carbon atom. The Hamiltonian is given by

    [math]\sum_{\langle i,j\rangle}a_i^\dagger a_j[/math]

    where the sum runs over all pairs of adjacent atoms and the a_i are fermionic annihilation operators. Intuitively, this says that the particles hop from one atom to a neighboring one. So for small timesteps, this interaction funtions like a partial SWAP gate with a small swapping angle. If we approximate the continuous time by discrete steps, we therefore obtain a three-dimensional network of partial SWAP gates, and these simulate the massless Dirac equation.

    Concerning the other issue, see my reply in your essay's forum.

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    • Post #47

    Tobias,

    Your essay is one of the better essays in the lot here. It was an enjoyable reading.

    Cheers LC

      Dear Tobias,

      I really love your idea of the graphene Dirac simulator. The second quantization in the tight-binding approximation to graphene that you give is really juicy! I will iimmediately explore this.

      We are currently communicating in parallel on our two blogs, and some of the ideas that I'm posting here are also reported in my reply to your last post on my blog.

      My problem is to prove that it is possible to simulate the Dirac equation by a quantum computer with a periodic topology of gate connections. This is also your problem, if your Graphene can be regarded as such a kind of a quantum computer (as you seem to assert in your answer). As you saw in my essay, I showed that this is possible in 1 plus 1 dimensions (with a mass-dependent renormalization of the speed of light). I'm trying now to prove it in 3 plus 1 dimensions (here it seems that a 5-simplex geometry is needed for each gate). Now, the problem is the following. In my blog you are mentioning a simple proof that a regular lattice will never give an isotropic propagation speed. How can you reconcile this with the covariance of the Dirac equation that you are simulating by the regular-lattice quantum-simulator graphene? I'm very intrigued and very curious.

      Let me compliment again on your work!

      Cheers,

      Mauro

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      • Post #49

      He is skilling indeed Tobias.A little of 3D harmonized with 4D and a little of rationality about the entanglement and it's very relevant.

      Now of course for a quantum computer , the realism is deterministic in the pure road of real numbers.The graphene is a step, a weak step.but it's well , they try to converge with the reality, it's the most important.

      Tobias....... operators hamiltonians and Laplacians more green and stokes more the rotational operators ....and if you insert the real number.....but perhaps an irrotational vectorial field is prefered U=-1/4INTdivVdv/r....poisson helping and the serie respected...and of course the harmonious function...a real puzzle all that ....fourier always is interesting.....now of course the volumes of entangled spheres is essential.....and what about the theory of big number and the probabilities and the errors also...Laplace where are you and Bernouilli....and the law of repartition of maxwell ...and pi always which smiles.....errors...moy. simple,moy. quadratic ,probable and precise...n=1/rac(pih)....DETERMINISM AND FINITE SERIE .....Pierce helping and Wolfram hihihi

      Spherically yours.

      steve

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      • Post #50

      Hi Tobias,

      Thanks for the introduction to graphene.

      Question, do you think graphene will show interference patterns similar to C60.

      I like your essay and think it is one of the best, but would encourage you to venture a little more into speculation. I think physics is at a local peak and it is going to be hard to get off it into something more productive without some leaps of faith.

      Don Limuti

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        • Post #51

        Tobias,

        Wow. I regret not reading your essay until now, but was happy I was able to rate it high before the close of the contest.

        Masterful job of bringing the essence of simulation and modeling theory down to Earth. Very nice explanation of the relation of the Dirac equation to spacetime values.

        I think you're overly modest about the significance of your explanation of continuous vs. discrete as relates to leading edge research. There are a number of important unsolved problems -- protein folding comes immediately to mind -- in which a continuous and random time dependent walk contrasts with the discrete lowest energy state. Classical computing hasn't been much help so far, that I know of; a simulation from another system could be a breakthrough.

        (I like the clever distinction between "quantum graphenity" and PI's "guantum graphity.")

        Hope you get a chance to read my essay. I, too, chose to survey the subject rather than dwell on research results. I think that your research program and mine have much in common mathematically, however.

        All best,

        Tom

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        • Post #52

        Your model is indeed interesting dear Narsep,

        Best

        Steve

        Hi Don,

        thank you for the feedback! Indeed I could have ventured more into speculation, but unfortunately at the present time I do not have interesting and original speculations worth mentioning...

        About the interference patterns: I'm not sure what you mean. C60 is a pretty small molecule which one can shoot at a diffraction grating and observe an interference pattern. At least in theory, this is not specific to a molecule of carbon atoms; it should work with anything of small enough size. As you probably know, the next step in this kind of experiments is to do it with a virus, which doesn't have anything to do with C60 or graphene. On the other hand, a graphene sheet can be quite large. So maybe you can elaborate on your question a little more?

        Hi there. I used this opportunity to write out some conditions for emergence of continuous structures, and in particular that of Lorentz invariance, in general based on the classification provided by effective field theory. Graphene famously has such emergent symmetry, but in more complicated models which include all the matter content and structures of the standard model, might be more difficult to achieve. If you are interested the argument (and known loopholes) are here:

        http://www.fqxi.org/community/forum/topic/856

        I am curious about your thoughts.

        Cheers,

        Moshe

        See the answer in your essay's forum! Essentially, the main point is that the emergence of the massless Dirac equation only holds for small particle momenta... For higher momenta, anisotropies appear also in graphene, and this is known as trigonal warping. These higher order contributions are suppressed by additional factors of the lattice spacing constant. So the momentum scale at which the anisotropies appear depends on the lattice spacing.

        Probably it should also be mentioned that this is why any extension of this to the massive Dirac equation is pointless. If the massless Dirac equation holds only for small momenta, then one can also approximate the system up to the same order by the Pauli equation, i.e. the Schrödinger equation with spin. No relativistic spacetime emerges in this case.

        Dear Tobias,

        Trigonal warping: very interesting! I like the idea of modified dispersion relation similar to those of Smolin, Magueijo and Amelino-Camelia. This may provide a way to discover the digital nature of reality!

        However, I cannot believe that the massless field has no digital analog, there must be a way! Otherwise we are proving that the world is not digital! In the reply to your thread in my blog I conjectured a possible mechanism to cure the problem with an anisotropic refraction index. I hope it will work, since I believe that "reality" is digital!

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        • Post #57

        Hi Tobias,

        I was thinking that it was possible to get single graphene rings and their interference would be "interesting".

        Good Luck,

        Don Limuti

        Tobias,

        I'm posting another question about your (let me say it again) very interesting work. I'm very interested in your graphene simulator, since, as you can imagine from my work, I want to understand more Dirac quantum simulation in space-dimensions d>1, e.g. your case d=2. The way in which I do things I have a tripartite gate, which indeed builds up a graphene spatial network, but it corresponds to a Dirac equations with a 3x3 (differential) Hamiltonian matrix, since the gate is tri-partite. I'm still trying to understand if this is the only possibility, but it looks so ... Now, I want to come back to your idea of the tight-binding effective Hamiltonian.

        The best way to explain myself, again, is through a figure. By the way, this is part of my talk at the March Meeting next tuesday. As you see, I'm quoting you!

          Dear Tobias,

          Congratulations on your dedication to the competition and your much deserved top 35 placing. I have a bugging question for you, which I've also posed to all the potential prize winners btw:

          Q: Coulomb's Law of electrostatics was modelled by Maxwell by mechanical means after his mathematical deductions as an added verification (thanks for that bit of info Edwin), which I highly admire. To me, this gives his equation some substance. I have a problem with the laws of gravity though, especially the mathematical representation that "every object attracts every other object equally in all directions." The 'fabric' of spacetime model of gravity doesn't lend itself to explain the law of electrostatics. Coulomb's law denotes two types of matter, one 'charged' positive and the opposite type 'charged' negative. An Archimedes screw model for the graviton can explain -both- the gravity law and the electrostatic law, whilst the 'fabric' of spacetime can't. Doesn't this by definition make the helical screw model better than than anything else that has been suggested for the mechanism of the gravity force?? Otherwise the unification of all the forces is an impossiblity imo. Do you have an opinion on my analysis at all?

          Best wishes,

          Alan

          First of all, I should point out again that the simulation of the Dirac equation by graphene has not been 'my' idea! So you may want to quote either Wallace, who first considered the tight-binding approximation in a two-dimensional hexagonal lattice, or Semenoff, who considered the simulation aspect already in 1984.

          Then I have to admit that I don't understand the correspondence between your model and the graphene lattice. In the latter, the gates are the edges of the hexagons, so they are bipartite. Anyway, I'm sure you have thought this over well, and I will understand the details in due time. Enjoy the Meeting :)