Discussion about the use of self-organisation for automatically "programming" networks of processing nodes.

Sunday, February 26, 2006

Quantum field = tokens in bins

I wonder whether some people find the phrase "quantum field theory" off-putting, especially when it is used in the context of networks that are discrete time dynamical systems, such as is described in my paper Discrete network dynamics. Part 1: Operator theory.

Quantum field theory is defined in wikipedia as:

Quantum field theory (QFT) is the application of quantum mechanics to fields. It provides a theoretical framework, widely used in particle physics and condensed matter physics, in which to formulate consistent quantum theories of many-particle systems, especially in situations where particles may be created and destroyed.

This is a physicists' definition of QFT, which ignores the wider use of the very useful algebraic properties of QFT in the description of many-particle systems. In the discussion below I will focus on bosonic statistics only, where the algebraic properties are defined using commutation rather than anti-commutation relations. Also, I will assume that the theory is defined on a discrete lattice, rather than on a continuous background space.

Stripped down to its bare essentials, QFT is a theory of how to manipulate "tokens" (e.g. particles) that are stored in "bins" (e.g. states). Each configuration of tokens-in-bins defines a particular set of bin occupancies (e.g. pure state).

Creation and annihilation operators applied to the occupants of these bins causes tokens to be created or annihilated in exactly the way that one would intuitively expect:
  1. A creation operator adds a token to a bin (there is only one way of doing this to a bin), whereas an annihilation operator removes a token from a bin (the number of ways of doing this is equal to the number of tokens in the bin). This corresponds to the commutation relation [ai, aj†] = δi,j.
  2. An annihilation operator completely erases a bin if it is already empty of tokens. This corresponds to annihilation of the vacuum state ai│0> = 0.
  3. A weighted sum of products of creation operators produces a correspondingly weighted sum of bin occupancies (i.e. mixture state).

The above description of the properties of creation and annihilation operators does not mention Planck's constant, nor does it specify what weights to use in a weighted sum of operators. The algebraic structure of QFT is not specifically designed for a quantum theory in which Planck's constant appears, and where the weights are complex-valued probability-amplitudes. Rather, stripped down to its bare essentials, QFT is a very convenient algebraic structure for doing the combinatoric book-keeping associated with the creation and annihilation of tokens in bins, where the weights are real-valued probabilities.

So the algebra of QFT is perfect for describing the manipulation of tokens in bins. Many of the algebraic techniques of QFT that have been developed in the context of physics can be reused in the context of tokens-in-bins. For instance, Feynman diagrams are used in physics to represent algebraic expressions that are formed from operators that generate interlinked sets of particle creation and annihilation operations. These diagrams can also be used to describe interlinked sets of operations on tokens in bins.

So quantum field theory (bosons on a lattice) is entirely appropriate for describing networks that are discrete time dynamical systems, in which the basic update operation consists of the creation and annihilation of tokens in bins.


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