Dear Ben
thank you very much for your careful reading of my paper and your appreciation. By the way, your post also attracted my attention to your essay, which I thus read carefully, and I'm going to report on your FQXi thread. As you have seen (and will better see after my reply), there are some strong common points between our two views, but also some relevant differences, about which I'll try to change your mind, since I'm very convinced about them. But on this, see your thread.
Before coming to your questions, I want to re-emphasize that my proposed new principles are already at some advanced stage, since so far I know the exact answer to many questions, and for many other ones I can foresee the route to answering them. What I can tell you is that the Dirac quantum automaton theory is almost finished in any dimension, and I'm going to put a set of two long papers on the web (one in collaboration with two my young collaborators Alessandro Bisio and Alessandro Tosini, and one with my elder collaborator Paolo Perinotti who just developed the theory for d=2,3). I have explored all scales, from the ultra-relativistic, to the Planckian-mass, and recovered correctly the usual Dirac theory in the "thermodynamical" limit (not the continuum limit!). We have already the leading terms for the Planck scale. For this I needed to develop a new dedicated asymptotic approach with Bisio and Tosini. One of the main thing that I have well understood is that the quantum nature of the causal network is crucial in recovering the Lorentz covariance. I like to epitomize this with the words: "a quantum digital universe".
But now, let's come to your questions:
Your POINT 1. I definitely agree with you that the principle of covariance, and all symmetries, are just approximate, and they are perfectly valid at scales above the Fermi's. What matters here is the Lorentz-more than the more general Poincare covariance, in the sense that homogeneity is inherent in the automaton description, and translation covariance/invariance are trivially recovered in the thermodynamic limit. Clearly, since all continuous symmetries are not true at the Planck scale, also all conservation laws must be rewritten and the digital form of Noether theorem should be given. But this is just a pleasant exercise in my case, and I hope I will find the time to doit in the next month. What I still don't know is what is the most general structure that is going to replace the Lie group of symmetry transformations, something more general of a discrete semigroup-and, I agree with you, this is a very relevant and fascinating problem. The quantum automaton gives a lot of phenomenology at all scales. All violations of symmetries can be seen already at the easiest level of dispersion relations, where it is a simple exercise to recover the Lorentz-covariant limit. Tosini is currently working in explicitly deriving the mechanism by which the Tomonaga rule (change of the quantum field state between infinitesimally close leaves in a foliation). But this is just a clever (and beautiful) exercise, as we already know that covariance is recovered, since we already recovered the Dirac quantum field theory.
Coming now to the case of homogeneous classical causal networks, in order to see what happens, you should take a look at my manuscript with Tosini [my reference [26] G. M. D'Ariano and A. Tosini, arXiv:1109.0118 (2011)]. With classical causal networks it is easy to understand what is the digital version of a foliation, and also to recover a digital version of Lorentz transformations. However, the "digital" nature of the clock at the Planck scale leads to a "coarse-graining" of events when boosting the frame, making the Lorentz transformation a semigroup-or else requiring explicitly the knowledge of the rest frame. But there is a stronger and easier-to-understand reason for not considering the classical causal network: just the fact that, as explained in my essay, you cannot recover the usual Minkowski metric space as emergent from event counting over the quantum network, due to the anisotropy of maximal speed of information flow, since as proven by Tobias Fritz the set of points that can be reached in a given maximum number of steps on a homogeneous network is always a polytope that does not approach a sphere in the limit of infinitely many steps (a version of the Weyl "tiling issue"). Finally, coming to the Sorkin approach-essentially the random version of my homogeneous network-I don't like it, for the following three reasons: 1) homogeneity in my case is the universality of the physical law. A random law needs another higher-level law that is non-random and which regulates the randomness (in my case is the quantum nature of the circuit). 2) Randomness where? We don't want a background where the random parameter is interpreted in terms of the metric ... 3) Where is quantum field theory? At least, in my case, I can have the Dirac field precisely emergent. But, maybe there is a conjunction logical link between the two approaches that I'm missing.
Your POINT 2. In my case the frame is only one (modulo translations): the one corresponding to the theory-e.g. Dirac's. You should understand that my and your viewpoints about causality are quite different. My network is not the usual causal network as that of Sorkin: it is a quantum automaton. Causality is a postulate of quantum theory, as established in my recent axiomatic work with Paolo Perinotti and Giulio Chiribella [[2] G. Chiribella, G. M. D'Ariano, P. Perinotti, Phys. Rev. A 84, 012311 (2011)]. Causality means independence of the probability distribution of state preparation from the choice of the following measurement (this is the same sense of Lucien Hardy's framework). Very shortly, this means that the causal links are the "wires" in the quantum circuit, whence they are fixed and they themselves establish the dynamical theory. I don't need to introduce a meta-dynamics for moving the wires. The theory is perfectly causal! I want to keep quantum theory, I don't want to modify it. Gravity must emerge as a thermodynamical effect a la Jacobson-Verlinde.
Your POINT 3. I agree with Sorkin in "order plus number equals geometry". But I disagree about the topological randomness (see end POINT 1). From what I understand from your paper, this is a strong common point of view between us.
Your POINT 4. My operational approach to QT is that of the above Ref. [2]. The problem is now recovering quantum field theory and the "mechanics" of the quantum, and this comes as emergent from the automaton. The automaton is perfect for the Feynman path integral: you have just a converging and well-defined path-sum over the quantum network of the automaton! But, as you will see on my comment to your essay, the global topology of the network is flat, e.g. it cannot be a torus: causal connection as a partial ordering IS transitive in my case, since I don't want to change quantum theory. Changing quantum theory in order to have time-loops, even for very long ranges, is a VERY difficult task, and we must to be well sure that we needed it before embarking on a new dead end!
Your POINT 5. I strongly believe in the Deutsch-Church-Turing principle. Richard Feynman declared many times that he believed it. There are many other motivations besides those that I expressed in my essay. It is not only the point that there are no divergencies and everything is computable. And that the path integral is well defined. And that we have a perfect match between experimental and theoretical protocols. But also the fact that computability is an unstable notion for infinite dimensions (proved by Arrighi). And more. In a scientific experimentable theory "infinite" is a "potential" notion, not "actual" one. Infinite-dimension must be needed only in the "thermodynamical" limit.
Thanks you very much for your stimulating questions. You are a "natural scientist" without prejudices, and hope we can discuss in person soon: it would make things much easier to explain!
Mauro
