Dirac's Classical-Quantum Analogy for the Harmonic Oscillator: Classical Aspects in Thermal Radiation Including Zero-Point Radiation

TL;DR

This paper explores the analogy between classical and quantum mechanics using Dirac's framework, focusing on the harmonic oscillator and the role of zero-point radiation in classical thermal equilibrium, highlighting connections across all temperatures.

Contribution

It demonstrates how classical systems in thermal radiation, including zero-point radiation, can exhibit structures analogous to quantum mechanics, extending Dirac's analogy to thermal contexts.

Findings

Classical thermal radiation includes zero-point radiation scaled by Planck's constant.

The harmonic oscillator shows classical-quantum correspondence at all temperatures.

Classical phase space distributions inherit quantum constants like rom thermal radiation.

Abstract

Dirac's Poisson-bracket-to-commutator analogy for the transition from classical to quantum mechanics assures that for many systems, the classical and quantum systems share the same algebraic structure. The quantum side of the analogy (involving operators on Hilbert space with commutators scaled by Planck's constant $\hbar$) not only gives the algebraic structure but also dictates the average values of physical quantities in the quantum ground state. On the other hand, the Poisson brackets of nonrelativistic mechanics, which give only the classical canonical transformations, do not give any values for physical quantities. Rather, one must go outside nonrelativistic classical mechanics in order to obtain a fundamental phase space distribution for classical physics. We assume that the values of physical quantities in classical theory at any temperature depend on the phase space probability distribution which arises from thermal radiation equilibrium including classical zero-point radiation with the scale set by Planck's constant $\hbar$. All mechanical systems in thermal radiation will inherit the constant $\hbar$ from thermal radiation. Here we note the connections between classical and quantum theories (agreement and contrasts) at all temperatures for the harmonic oscillator in one and three spatial dimensions.