Classical Evolution Without Evolution

TL;DR

This paper demonstrates how classical physics can exhibit dynamics without explicit evolution by using a formalism similar to quantum mechanics, emphasizing energy conservation and correlations between subsystems.

Contribution

It extends the Page-Wootters argument from quantum to classical physics using a Hamilton-Jacobi-like formalism, highlighting the role of conserved quantities in timeless dynamics.

Findings

Classical dynamics can be derived without explicit evolution using a formalism akin to quantum theory.

Energy conservation leads to correlations that produce effective dynamics in classical systems.

The classical approach differs from quantum in the interpretation of mixed states and the role of the higher Hilbert space.

Abstract

The well known argument of Page and Wootters demonstrates how to "derive" the usual quantum dynamics of a subsystem in a global state which is an eigenstate of the total Hamiltonian. I show how the same argument can be made in classical physics, by using a formalism that closely resembles the quantum one. This is not surprising since the Hamilton-Jacobi formulation of classical dynamics is precisely motivated by the logic of timeless dynamics. Ultimately, the key to obtaining dynamics without dynamics is the principle of energy conservation which leads to correlations between times pertaining to different subsystems. The same can, of course, be said about all other conserved quantities and we show how to address this problem in its full generality so as to realise rotation without rotation, translation without translation and so on. The classical and quantum interpretations do, however, have one major difference and this is the Church of the Higher Hilbert Space interpretation of mixtures, which only exists in quantum physics. We discuss a few consequences of this point.