Quantum theory from classical mechanics near equilibrium

TL;DR

The work addresses whether quantum theory can be derived from classical mechanics when observers are restricted to quadratic, near-equilibrium observables, realized through complex coordinates $z_a=\frac{1}{\sqrt{2}}(q^a+ip_a)$. Using these coordinates, it constructs an unnormalized density-like operator $K_{ab}=\int \bar{z}_a z_b\, \rho\, d\bar{z}dz$ and shows that quadratic observables map to Hermitian operators $\hat{C}$ with expectation values $\mathrm{Tr}(\hat{C}\hat{K})$. The dynamics reduce to quantum-like evolution $i\frac{d\hat{C}}{dt}=[\hat{C},\hat{h}]$ under a quadratic Hamiltonian $H(\bar{z},z)=h^{ab}\bar{z}_a z_b$, and decoherence arises from adiabatic perturbations, leading to normalization to a density matrix with an effective $\hbar=c^{-1}$. Geometrically, the construction fits a moment-map framework on a symplectic/Poisson manifold, and the results imply that low-energy classical systems with finite degrees of freedom naturally exhibit conventional quantum mechanics, drawing connections to second-quantization.

Abstract

We consider classical theories described by Hamiltonians $H(p,q)$ that have a non-degenerate minimum at the point where generalized momenta $p$ and generalized coordinates $q$ vanish. We assume that the sum of squares of generalized momenta and generalized coordinates is an integral of motion. In this situation, in the neighborhood of the point $p=0, q=0$ quadratic part of a Hamiltonian plays a dominant role. We suppose that a classical observer can observe only physical quantities corresponding to quadratic Hamiltonians and show that in this case, he should conclude that the laws of quantum theory govern his world.