Tldr is that the Koopman operator evolves functions of the dynamical variables; it’s an infinite dimensional linear operator, thus it has eigenfunctions and eigenvalues. Conserved functions of state have zero-eigenvalue; nearly conserved quantities have ‘small’ eigenvalues, corresponding to a slower time rate of decay. Then you can characterise all functions in terms of the decay rate of their self-correlation. So the dominant koopman eigenvalue of a (not necessarily eigenfunction) is a scalar value corresponding to how “not conserved” it is.
You can also look at the Jordan blocks, which correspond to groups of functions which are predictively closed (you can predict their future values by knowing only the present values of those functions). These are some threads I’ve found interesting in thinking about natural ontology but I do not think they are sufficient—for example they give a very clear picture of intra-realization correlations, but do not have a good way of talking about inter-realization correlations or sensitivity to initial conditions.
The dynamical systems way of framing this I like the most is Koopman analysis; Steve Brunton has a great family of talks on on youtube: https://www.youtube.com/watch?v=J7s0XNT96ag
Tldr is that the Koopman operator evolves functions of the dynamical variables; it’s an infinite dimensional linear operator, thus it has eigenfunctions and eigenvalues. Conserved functions of state have zero-eigenvalue; nearly conserved quantities have ‘small’ eigenvalues, corresponding to a slower time rate of decay. Then you can characterise all functions in terms of the decay rate of their self-correlation. So the dominant koopman eigenvalue of a (not necessarily eigenfunction) is a scalar value corresponding to how “not conserved” it is.
You can also look at the Jordan blocks, which correspond to groups of functions which are predictively closed (you can predict their future values by knowing only the present values of those functions). These are some threads I’ve found interesting in thinking about natural ontology but I do not think they are sufficient—for example they give a very clear picture of intra-realization correlations, but do not have a good way of talking about inter-realization correlations or sensitivity to initial conditions.