Suppose we take Vicary's quantum harmonic oscilator as a kind of "toy quantum field theory". Next, take the category of internal comonoids to not represent the background causal structure. We normally take the comonoids to be something like the cobordisms in TQFT. Instead, we take the internal comonoids to represent classical causal structure (where ordering is taken from the string diagrams in the category of internal comonoids). This classical causal structure can be understood as the ordering of events in your laboratory (turning knobs, flashing lights etc.). The background, or underlying causal structure, is taken as the string diagrams in the base monoidal category. The idea being that the underlying causal structure is quantum and the interface to it is via the string diagrams in the classical surface which we see as the category of internal comonids. Thus, looked at this way, a quantum field theory might be seen as an epistemic restriction on (quantum) causal structure. We map diagrams in the background, quantum causal structure into the classical interface and this is the concept of an epistemic restriction. I guess the question is, can we interpret quantum field theory as an epistemic restriction on causal structure?

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    $\begingroup$ Maybe, but I think you have to make detailed contact either with Wightman QFT, Lagrangian QFT, or experiment to justify a category theory approach. The space-time metric structure of QFT makes it more than a system of QHOs. As well as the obvious microcausality constraint on the algebra of observables and the requirement of a relativistic transformation law, there is also the cluster decomposition constraint on the state. I don't see that Vicary gets you close enough. In any case, your final question is more-or-less disjoint from your discussion of a category theory approach. $\endgroup$ – Peter Morgan Jul 9 '12 at 18:29
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    $\begingroup$ I am likely misunderstanding your final question, but I have to confess that it feels a bit tautological in this sense: Isn't any meaningful theory of physics, however it may be represented mathematically, necessarily also an epistemic restriction on causal structure? If not, what would be a counter example? $\endgroup$ – Terry Bollinger Jul 10 '12 at 2:51
  • $\begingroup$ I will quickly respond to Terry's comment. In fact, deriving any kind of physics from an epistemic restriction is very hard. That is why Spekken's Toy Model was so important and unique. $\endgroup$ – Ben Sprott Jul 13 '12 at 1:39
  • $\begingroup$ I suppose stat mech leading to thermodynamics is another example. $\endgroup$ – Ben Sprott Jul 13 '12 at 2:21

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