An Operational Representation of Systems: A Worked Construction in Lattice Gauge Theory

This paper introduces a minimal, ordinal representation of systems and tests it in lattice gauge theory.

The representation takes a system to be comprised of three things:

• something that drives it,
• something that constrains it,
• and the system’s own way of holding the two together.

The representation is minimal because it uses only those three roles.

The smallest part of a system, which we call an element, divides what reaches it between what it passes on and what it holds back. Only those two shares can be observed, and the representation is ordinal because it carries nothing but shares of a unit, and no absolute scale. The axes against which they are measured cannot, and each element has axes of its own, so a rule for carrying a state from one element to the next has to be supplied.

That is also the shape of lattice gauge theory: sites whose local frames no measurement can see, links that carry a rule between neighbouring frames, and observables that survive being carried round closed loops. The paper states the systems representation as axioms and shows that the theory’s structures follow from them and from assumptions that are named: that relations act transitively, that amplitudes are real or complex, and a chosen form of the dynamics and of the loss round a loop. The rule on a link is an element of a group; that the relations form a group is one of the axioms, and which group it can be is then derived. The invisibility of frames is the gauge freedom, and it gives Gauss’s law. The loss of carrying a state round a loop, averaged over the state, is the Wilson action, and whether the amplitudes are real or complex, and which group acts, fixes the representation it is written in. Keeping the states as variables fixes an interaction between neighbouring loops, which is checked by direct diagonalisation.

The aim is to show that a representation of systems in general can be made precise enough to construct and check definite structure in an exacting theory, and that the theory then becomes intelligible as the consequence of a few statements about systems.

Version 1.2: The abstract's account of why the representation is ordinal reworded for readability; no change of substance.