Physics 151 (Fall 2023): KvN theory

The motivation for this final project is the adjoint problem of Hamiltonian mechanics: understanding how the evolution of a subsystem can be used to infer properties of the composite system. The goal is characterizing dynamical closures—the extension of subsystem dynamics into the Hamiltonian dynamics of a larger system. This problem is not solved by this project. However, the steps towards solving this problem provided insights into the relation between kinematics and dynamics, as well as the boundary between classical and quantum theories.

Main takeaways:

1. Koopman von Neumann theory: reinterpreting classical dynamics within Hilbert spaces to explore analogies with quantum systems; this perspective highlights the role of noncommutative observables in quantum theory.
2. Stinespring's dilation theorem: establishes the feasibility of snapshot closures for classical systems, using the similar result from quantum theory (at the cost of dilating to \(n^2\) degrees of freedom); the remaining problem is assembling possible dilations into a canonical transformation.
3. Numerical exploration of the coupled harmonic oscillator, and intuition into how the dynamics of one oscillator relates to the full system’s evolution.