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arXiv:2603.13446

Classical linear oscillator in classical electrodynamics with classical zero-point radiation

Timothy H. Boyer

physics.class-phquant-phSubmitted Mar 13, 2026• Announced Mar 16, 2026
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Abstract

A classical linear oscillator is treated in the small amplitude limit so that it will be approximately relativistic. The oscillator involves a charge particle in a linear potential in classical zero-point radiation. It is found that the ground state is energy balanced with the power lost in radiation emission equal to the average power gained from resonance with the classical zero-point radiation. Also the oscillator is found to have resonant excited states where the energy emitted as dipole radiation is balanced on average by the energy gained from the zero-point radiation when the action variable of the mechanical system is given by J=(n+1/2)(h/2pi).

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31 pages

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A classical linear oscillator is treated in the small amplitude limit so that it will be approximately relativistic. The oscillator involves a charge particle in a linear potential in classical zero-point radiation. It is found that the ground state is energy balanced with the power lost in radiation emission equal to the average power gained from resonance with the classical zero-point radiation. Also the oscillator is found to have resonant excited states where the energy emitted as dipole radiation is balanced on average by the energy gained from the zero-point radiation when the action variable of the mechanical system is given by J=(n+1/2)(h/2pi).
Classical electrodynamics including classical electromagnetic zero-point radiation leads to a ground state and resonant excited states for a charged particle in a Coulomb potential. These resonant states correspond to integer values of the action variables analogous to those appearing in the Bohr-Sommerfeld theory of the hydrogen atom. The work on classical zero-point radiation reported here is a continuation of the analysis reported in 1975, but with the addition of the ideas of relativity and resonance between the charged-particle orbit and classical zero-point radiation.
The theory of absorption and emission of electromagnetic radiation by an oscillator consisting of the atomic nucleus and one electrically charged particle is deduced using classical electrodynamics. In the steady state of an atom, emission and absorption of electromagnetic radiation are equal, so the atom is stable. In order to include reactive effects of electromagnetic radiation in the motion equations, the Newton equation is modified by adding the radiative reaction force. This paper is an introduction to the derivation of the basic assumptions of quantum mechanics.
Abstract This paper is a study of a statistical ensemble of classical harmonic oscillators which is stationary in time, and whose position and momentum distribution functions are those of the corresponding quantum-mechanical oscillator in its ground state. If the oscillating particle is charged, then in order to maintain the distribution stationary a certain random electromagnetic field must be present. The intensity distribution of the radiation field is calculated, and it is found to be identical with the ‘photon vacuum’ of quantum electrodynamics. It is suggested, therefore, that this radiation field, which in quantum field theory is treated entirely formally, might exist in the classical sense. The method is extended to the excited states. It is found that the classical ensembles corresponding to these have probability distributions which may be negative. However, when attention is shifted to the quantum-mechanical ‘mixture’, this is no longer the case. Furthermore, the intensity distribution of the radiation field at temperature T is shown to be simply the sum of the Planck distribution and the previously obtained zero-temperature field. The application of these results to statistical mechanics is discussed. In this paper the treatment is non-relativistic throughout, but radiative corrections of the type which give rise to the Lamb shift are an integral part of the theory.
The present paper is devoted to the detailed study of quantization and evolution of the point limit of the Pauli-Fierz model for a charged oscillator interacting with the electromagnetic field in dipole approximation. In particular, a well defined dynamics is constructed for the classical model, which is subsequently quantized according to the Segal scheme. To this end, the classical model in the point limit, already obtained by Noja and Posilicano [Ann. I.H.P. Phys. Theor. 71, 425 (1999)], is reformulated as a second order abstract wave equation, and a consistent quantum evolution is given. This allows a study of the behavior of the survival and transition amplitudes for the process of decay of the excited states of the charged particle, and the emission of photons in the decay process. In particular, for the survival amplitude the exact time behavior is found. This is completely determined by the resonances of the systems plus a tail term prevailing in the asymptotic, long time regime. Moreover, the survival amplitude exhibits in a fairly clear way the Lamb shift correction to the unperturbed frequencies of the oscillator.
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 ℏ) 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 that arises from thermal radiation equilibrium including classical zero-point radiation with the scale set by Planck's constant ℏ. All mechanical systems in thermal radiation will inherit the constant ℏ 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.