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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 :)

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        • Post #63

        http://prl.aps.org/abstract/PRL/v106/i11/e116803

        Tobias,

        in a quantum circuit there are both space and time. If you consider the graphene as a quantum circuit (namely the gates are the edges of the hexagon, whence they are bipartite), then your computational circuit is 2dl, means there is only a single space dimension! Then my simple circuit simulates the Dirac in 1+1 very well, and graphene would not. I think that you should look at graphene as a the spatial projection (a leaf in the rest-frame foliation) of a 2+1 dim. circuit!

        Cheers

        I see, so by "circuit" you mean not just the qubits themselves, but the qubits together with the gates as a circuit in spacetime. Yes, this makes sense, given that one usually draws a quantum circuit as a two-dimensional figure with one direction being space and the other direction being time.

        Am I understanding correctly that the hexagonal lattice is split up into two layers, corresponding to adjacent spatial slices? I am a bit confused about this point, since a qubit at every instant of time and have a 1-dimensional worldline, shouldn't it?

        Dear Tobias,

        it seems to me that electrons flowing in the graphene may simulate vs time the evolution of the Dirac qstate in 2dspace, but it is certainly not the quantum circuit that simulates 2plus1 Dirac.

        I also need to read the original paper that you are quoting.

        Second: the way in which I see things, the Dirac eq. in 2plus1dims should have a 3x3 Ham matrix since the gate is tri-partite---not a 2x2 as you write in your paper. And my way I automatically get the spin in 3plus1dims.

        Reader of this blog: look for figure Dirac2plus1_small.jpg in the next thread.

        Cheers