This was the worst feeling: nearly a whole year gone with absolutely nothing to show for it. Worse, I was wracked with guilt, feeling I’d totally wasted Nick’s time.

…

Things soon got better. By a miracle my postdoc had been extended for a while, so at least I wasn’t on the job market straight away. Secondly, at the end of 2004 I went to a truly inspiring conference at the Isaac Newton institute where I met up with Guifre Vidal who showed me something amazing: the Multiscale Entanglement Renormalisation Ansatz, and I realised that what I should do is focus on more my core skill set (quantum entanglement and many body quantum spin systems). I began working on tensor networks, read a fantastic paper of Hastings, and got into the Lieb–Robinson game.

If I had my time again what would I do differently? I don’t regret working on this problem. It was worth a try. My mistake was to keep working on it, to the exclusion of everything else, for too long. These days I am mindful of the advice of Wheeler: you should never do a calculation until you already know the answer. I also try to keep a portfolio of problems on the go, some risky ones, and some safe ones. (More on that in a future post.) Finally, upon reflection, I think my motivation for working on this problem was totally wrong. I was primarily interested in solving a famous problem and becoming famous rather than the problem itself. In the past decade I’ve learnt to be very critical of this kind of motivation, as I’ve seldom found it successful.

]]>In the signal flow diagrams of control theory, as explained in my talk, the edges are labelled with spaces, while the vertices are labelled by relations from a product of spaces to another product of spaces. However, in the version of the theory I describe here, all these spaces are *the same!* They’re all the same space of ‘signals’, a vector space over the field of rational functions $\mathbb{R}(s).$ where acts as So, the labelling of edges is invisible in this example. It becomes visible in more general kinds of signal flow diagrams where we have different types of signal.

(And again, it’s a good sign that ‘type’ here is being used the way they do in computer science, meaning an object in a category.)

I believe the spaces I’ve just been talking about should be viewed as configuration spaces rather than phases spaces. As you’ll see near the end of that talk, when we get signal flow diagrams from electrical circuit diagrams the signals come in pairs: the voltage and current along a wire. These pairs should be thought of as lying in and the symplectic structure on become important.

Thanks for prodding me: I had to have some new ideas to make up this explanation, and I’ll have to do more work to flesh it out! But the simple point is: if you want a network that “has phase spaces”, you’re going to take a graph and label its edges and/or vertices with phase spaces.

]]>We can also study the behavior of a large network by looking at the maps into it from other networks and maps out of it to other networks. I think you would agree with that.

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