### Discrete network dynamics. Part 1: Operator theory (update 2)

I need to update what is going on with my paper "Discrete network dynamics. Part 1: Operator theory", which can be found in arXiv at cs.NE/0511027. Yes, it was that long ago and the paper is

*still*not published, nor do I know

*how*I can get it published. I previously blogged about this here, here, here, and here.

I have been getting rather evasive feedback from various people about this paper, and I don't like the sound of what I hear at all. In a nutshell, there is a widespread view that my use of quantum field theory is misguided and/or wrong.

One critic said that I didn't understand quantum mechanics, which I thought was rather odd given that my PhD is in quantum chromodynamics! That comment immediately told me that he had

*not*read my paper, or at least had not read it very carefully, because I

*don't*actually use quantum mechanics in the paper, as least not in the sense that physicists use it (i.e. the sense in which the criticism was made). In my paper I explain how I use operators to implement the elementary processes of adding samples to and removing samples from histogram bins, and that these operators have

*exactly*the same algebraic properties as the bosonic creation and annihilation operators that are used in physics. To this extent, what I am doing is

*algebraically*equivalent to a quantum field theory, and I explained this in greater detail in an earlier posting here. What I am certainly

*not*doing is using a representation of these operators in which they act on an underlying wave function, because there

*isn't*any wave function defined in my samples-in-histogram-bins model. My creation and annihilation operators should be thought of as little pieces of algorithm whose effect is to add and remove samples from histogram bins.

All I use is operators for creating and annihilating quanta; the model is defined at the level of these quanta, and there is no deeper level of theory. This is where the QFT that I use is a

*subset*of the type of QFT that physicists are familiar with; I make use of only the combinatoric properties of field quanta, which are elegantly summarised in the algebraic properties of the corresponding creation and annihilation operators. I also use a lattice QFT, rather than a continuum QFT, but I assume that the difference between these two is relatively benign (

*pace*, mathematical purists!). Because physicists are exposed to QFT in (relativistic) particle physics and in (non-relativistic) condensed matter physics, they naturally

*assume*that QFT needs an underlying wave function with the usual quantum mechanical behaviours that they have been taught about. This is a very narrow definition of QFT, because you can have a QFT whose quanta are built in any way that you want.

One can manipulate these creation and annhilation operators by using their algebraic properties to rearrange "operator products" in various ways, and thus break operator expressions apart into a sum of contributions with different combinatoric properties, each weighted by a combinatoric factor that is automatically generated by the algebraic manipulations. These manipulations have the same general structure as the sorts of manipulation that occur in "operator product expansion" calculations in high energy physics; my PhD dissertation is full of this sort of calculation, and some relevant papers that I contributed to can be found by doing a search of the SPIRES database here.

All the above leaves me no choice but to describe what I write about in my paper "Discrete network dynamics. Part 1: Operator theory" as a "quantum field theory".

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