Dear Ben
I enjoyed reading our manuscript very much! You made the point about the current situation very clear and synthetic, and I agree on many of your points. As you will see in my answer to your post on my essay http://fqxi.org/community/forum/topic/1506, even though there are some strong common points between our two manuscripts, there are also some relevant differences, about which I'll try to change your mind.
POINTS ON WHICH WE AGREE.
I agree on inexactness of symmetries and covariance, and that they should hold only at the Fermi scale and above. On the quantum automaton theory, this is exactly the case.
Lorentz covariance (more generally Poincare covariance, but translations are almost trivial, since homogeneity is inherent in the automaton description) are recovered in the "thermodynamic" limit of infinite automaton steps and systems. 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. The most general structure that is going to replace the Lie group of symmetry transformations, I agree is likely to be something more primitive than a group, I can say that in my case is likely to be a discrete semigroup, that is approximated by a Lie group at the Fermi scale. In the automaton, all violations of symmetries can be seen already at the easiest level of the dispersion relations.
We (and everybody should) agree that between two theories explaining the same physics, "parsimony" and "more possibilities for falsification" should be taken as the main motivations in choosing one of the two. In my cellular automaton approach parsimony comes from taking Quantum Theory as the only very fundamental theory. Relativity is emerging. GR must come out as the description of an emergent gravity in the "thermodynamic limit" a la Jacobson-Verlinde. Clearly for falsifiability we need experiments at the Planck scale, e.g. the Craig Hogan's [Scientific American, feb. 2012] (really very nice experiment: I visited his lab).
Background independence of the theory and physics as emergent: out of question!
We need to incorporate scale-dependence in the theories: this is already the case of Planck scale incorporation!
Proof is lacking that antimatter interacts in the same way as ordinary matter gravitationally: right! Something frequently forgotten!
Spacetime is not a manifold: out of question! And most likely it is not commutative (I hope to recover this from the Dirac automaton in 3d).
We need to reinterpret the principles of causality and covariance, and covariance should be viewed in order-theoretic terms. Agreed, but my solution is different from the one that you propose.
POINTS ON WHICH IT SEEMS THAT WE DISAGREE
I think that your causal metric hypothesis in some way is related to my quantum causality Deutsch-Church-Turing principle, i.e. in short the quantum automaton.
But my notion of causality I think is very different from yours! And, is more similar to the canonical notion. The disagreement is that my causality is definitely transitive and acyclic! It is also countable (discreteness comes from the requirement of distinguishing cause and effect: sees definition) and locally finite (from the Deutsch-Church-Turing principle!) Why I want a transitive and acyclic causality? Because I don't want to modify Quantum Theory! 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 in the usual sense! I want to keep quantum theory: gravity must emerge as a thermodynamical effect a la Jacobson-Verlinde.
You say that "intransitivity of the binary relation generating the causal order is self-evident at large scales": where?? We have no evidence at all. We believe in General Relativity, and take any astrophysical observation as an evidence of the theory. I want a direct evidence! I understand that you want to give up acyclicity (whence transitivity) for keeping GR, but this is not an experimental motivation.
"Metric" properties of space-time unfortunately involve an additional information besides the binary relation generating the causal order, and this is the fact that the causal relation is of quantum nature: is a quantum interaction. One of the main thing that I have well understood is that we live in "a quantum digital universe": the quantum nature of the causal network is crucial in recovering the Lorentz covariance. I explained more in my reply to your post on my essay. The scale factor definitely must come from the Planck distance.
Dimension is an emergent property? I'm not sure. If you believe in causal networks, the graphs dimension of the network (which equals the dimension of the emerging space-time)
depends on the topology of causal connections. These ARE the theory. Having dimension as emergent would correspond to have the most basic theory as emergent. Causality is not emergent: causality is our way of describing physics. Moreover, let me comment on your apparent connection with the Feynman path-integral. The closed trajectories in the Feynman integral have nothing to do with acyclicity of causality, since the fact that you can evaluate the probability amplitude of coming back to the same state doesn't mean that the evolution is cyclic.
Finally: do systems evolve with respect to an independent "time parameter". Time is emergent, and time in the usual Einstein sense need a synchronization procedure. But Lorentz covariance emerges from the automaton, and there we have an independent "discrete time parameter" which is just the number of unitary steps of the automaton!
Thank you again for your essay! I really liked it a lot!
I hope to meet you soon for the easiness and the pleasure of discussing in person.
Giacomo Mauro D'Ariano
