arXiv subjects

Timothy H. Boyer

Publications and source records attributed to Timothy H. Boyer.

At least 19 recordsLinked to original sources

arXiv2026

Conformal Symmetry and Planck's Thermal Radiation in Classical Physics

Thermal radiation is discussed for a classical scalar field. Dilation and acceleration appear in conformal symmetry, and so are treated in connection with zero-point radiation and equilibrium thermal radiation. Classical zero-point radiation is completely invariant under dilation in a inertial frame. However, in a Rindler frame, zero-point radiation takes a different form which is related to Planck's thermal spectrum with temperature T related to acceleration. Traditionally, zero-point radiation is said to take a thermal (Unruh) form in a Rindler frame. However, the classical viewpoint indicates that zero-point radiation in a Rindler frame requires a divergent frequency spectrum and so is not thermal radiation. The Unruh form suggests an additional classical thermal spectrum satisfying the Stefan-Boltzmann relation.

physics.class-ph↗

arXiv2026

Hydrogen Molecular Ion and Molecule in Classical Electrodynamics with Classical Zero-Point Radiation

The hydrogen molecular ion and the hydrogen molecule are treated in an approximate calculation based on classical electrodynamics which includes classical zero-point radiation. It is found within the classical theory that a molecular ion is less-well bound than a hydrogen atom plus a distant proton. The smaller binding energy is explained due to the repulsive nature of the force between the proton and the atom when considering the most natural resonant orbit of the electron. The approximate classical electromagnetic calculation gives a binding energy of $2.1eV$ and a proton separation of $0.916A^{o}$ for the hydrogen molecular ion. The hydrogen molecule is formed by adding a single additional electron to the ion. The electrostatic attraction of the electron to the ion gives a binding energy of $4.6eV$ and a inter-proton separation of $0.6A^{o}$ for the hydrogen molecule in this classical electromagnetic approximation.

physics.class-ph↗

arXiv2026

Helium Ground State Treated in Classical Physics with Classical Zero-Point Radiation

The ground state of the helium atom is considered within classical electrodynamics which includes classical electromagnetic zero-point radiation. Approximate energy balance between energy loss through emitted radiation and energy gain from classical zero-point radiation is found when the two electrons are treated as charges located on opposite ends of a diameter of a common circular orbit around the nucleus. The classical result gives approximately the same value as that given by the quantum calculation, but is a different value.

physics.atom-ph↗

arXiv2026

Magnetic Field Applied to the Classical Hydrogen Atom Treated in Classical Electrodynamics with Classical Zero-Point Radiation

An external magnetic field is applied to the classical hydrogen atom treated in classical electromagnetic theory including classical electromagnetic zero-point radiation. In an earlier article, it was shown that the average value of each of the electron's action variables appears as a discrete value, corresponding to a representation of the rotation group, because of resonance between the periodic orbit of the electron and the random classical zero-point radiation. Here it is shown that, in the presence of a magnetic field and because of resonance, the classical orbital motion of the electron is in resonance with random classical zero-point radiation only for orientations of the orbit which take integer values for the angle made with the direction of the magnetic field, but excluding the m=0 orientation where the magnetic field is parallel to the orbital plane of the electron. Classical electromagnetic explanations are given for the Stern-Gerlach result, and for the Zeeman effect.

physics.class-ph↗

arXiv2026

Criterion for the Thermal Radiation Spectrum in Classical Physics

Two criteria for the spectra of relativistic waves are proposed. Zero-point radiation provides the identity representation of the conformal group in Minkowski spacetime. Thermal radiation provides the irreducible representation of the conformal group in Minkowski spacetime which involves exactly one scaling parameter (the temperature) which is also time-stationary in a Rindler frame. Zero-point radiation is the limit of thermal radiation as the temperature goes to zero. Crucially, both zero-point radiation and thermal radiation take basically the same functional form in a Rindler frame. For relativistic scalar waves, a full derivation of the Planck spectrum including zero-point radiation is obtained with the classical theory.

physics.class-ph↗

arXiv2026

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

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).

physics.class-ph↗

arXiv2026

Relativistic hydrogen in classical electrodynamics with classical zero-point radiation

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.

physics.class-ph↗

arXiv2026

Zeeman effect in hydrogen treated in classical physics with classical zero-point radiation

The Zeeman effect for the low resonant energy states of hydrogen is treated with classical electrodynamics including classical zero-point radiation. The electron is regarded as a classical charged particle in a Coulomb potential. The "space quantization" of old quantum theory, the Sommerfeld relativistic result, and the Stern-Gerlach experiment are all considered.

physics.class-ph↗

arXiv2024

Symmetries and Thermal Radiation: A Classical Derivation of the Planck Spectrum

A derivation of the Planck spectrum for thermal radiation is given based upon wave fluctuations within relativistic classical physics. The derivation depends crucially on thermal fluctuations existing above the fundamental inertial-frame-independent fluctuations of classical zero-point radiation. Such frame-independent zero-point fluctuations exist only in a relativistic wave theory and cannot exist in a nonrelativistic wave theory. Thus such a classical derivation of the Planck spectrum exists in a Lorentz-covariant classical theory, such as classical electrodynamics, but not in a Galilean-covariant theory where all waves are based upon material media. Classical zero-point radiation provides a purely classical alternative to quanta in the analysis of the Planck spectrum.

hep-ph↗

arXiv2024

Point-Charge Models and Averages for Electromagnetic Quantities Considered in Two Relativistic Inertial Frames

Electromagnetic quantities at a spacetime point have tensor Lorentz transformations between relatively-moving inertial frames. However, since the Lorentz transformation of time between inertial frames depends upon both the time and space coordinates, averages of electrodynamic quantities at a single time will in general depend upon the inertial frame, and will differ between inertial frames. Here we illustrate how the use of continuous charge and current distributions rather than point-charge distributions can lead to physically mystifying and even inaccurate results for electromagnetic quantities and physical phenomena. The discrepancy noted between the average electric field values in different inertial frames is particularly striking because it is first order in the relative velocity between the frames.

physics.class-ph↗

arXiv2023

A Charged Particle Must Be Treated Relativistically in Classical Theory

A charged particle which is allowed to accelerate must have relativistic behavior because it is coupled to electromagnetic radiation which propagates at the speed of light. We treat the simple steady-state situation of a charged particle moving in a circular orbit with counter-propagating plane waves providing the power which balances the energy radiated away by the accelerating charge. It is emphasized that only an electromagnetic arrangement, such as a Coulomb potential or a constant magnetic field, can provide the relativistic central force for the particle motion.

physics.class-ph↗

arXiv2023

Concerning the Direction of the Aharonov-Bohm Deflection

The interaction of a solenoid with a passing charged particle can be treated within classical or quantum physics. If charged particles pass around both sides of a solenoid, there is an experimentally-observed Aharonov-Bohm deflection of the double-slit particle interference pattern between charges passing on opposite sides. Such a deflection can be obtained by a classical force calculation. Although the magnitude of the angular deflection agrees between the classical force calculation and the quantum topological theory, the direction of the predicted deflection is opposite. Here we point out the simple basis for the direction of the deflection based upon classical electrodynamics and based upon quantum theory, and we mention analogues, both the electrostatic deflection of the particle interference pattern and the optical analogue of the classical calculation. The deflection direction involves an experimental question which is addressed rarely if ever.

quant-ph↗

arXiv2023

Classical Electromagnetic Interaction of a Charge with a Solenoid or Toroid

The Aharonov-Bohm phase shift in a particle interference pattern when electrons pass a long solenoid is identical in form with the optical interference pattern shift when a piece of retarding glass is introduced into one path of a two-beam optical interference pattern. The particle interference-pattern deflection is a relativistic effect of order 1/c^2, though this relativity aspect is rarely mentioned in the literature. Here we give a thorough analysis of the classical electromagnetic aspects of the interaction between a solenoid or toroid and a charged particle. We point out the magnetic Lorentz force which the solenoid or toroid experiences due to a passing charge. Although analysis in the rest frame of the solenoid or toroid will involve back Faraday fields on the charge, the analysis in the inertial frame in which the charge is initially at rest involves forces due to only electric fields where forces are equal in magnitude and opposite in direction. The classical analysis is made using the Darwin Lagrangian. We point out that the classical analysis suggests an angular deflection independent of Planck's constant where the deflection magnitude is identical with that given by the traditional quantum analysis, but where the deflection direction is unambiguous.

physics.class-ph↗

arXiv2023

The Classical Aharonov-Bohm Interaction as a Relativity Paradox

The situation of a charged particle passing down the symmetry axis through a magnetic toroid presents a relativity paradox; different inertial frames suggest different forces on the charge and on the toroid due to the unperturbed systems. We review the charge-toroid interaction and suggest that the magnetic Aharonov-Bohm situation is misunderstood because of unfamiliarity with the acceleration fields following from the Darwin Lagrangian, which go unmentioned in recent textbooks of classical electromagnetism.

physics.class-ph↗

arXiv2022

Concerning Classical Forces, Energies, and Potentials for Accelerated Point Charges

Although the expressions for energy densities involving electric and magnetic fields are exactly analogous, the connections to forces and electromagnetic potentials are vastly different. For electrostatic situations, the changes in the \textit{electric} energy can be related directly to \textit{electric} forces and to the electrostatic potential. The situation involving magnetic forces and energy changes involves two fundamentally different situations. For charged particles moving with constant velocities, the changes in both electric and magnetic field energies are provided by the external forces that keep the particles' velocities constant; there are no Faraday acceleration electric fields in this situation. However, for particles that change speed, the changes in \textit{magnetic} energy density are related to acceleration-dependent Faraday \textit{electric} fields. Current undergraduate and graduate textbooks deal only with highly symmetric situations where the Faraday electric fields are easily calculated from the time-changing magnetic flux. However, in situations that lack high symmetry, such as the magnetic Aharonov-Bohm situation, the back (Faraday) acceleration electric fields of point charges seem unfamiliar. In this article, we present a simple unsymmetric example and analyze it using the Darwin Lagrangian. In \textit{all} cases involving changing velocities of the current carriers, it is the work done by the back (Faraday) acceleration \textit{electric} fields that balances the \textit{magnetic} energy changes.

physics.class-ph↗

arXiv2021

Disguised Electromagnetic Connections in Classical Electron Theory

In the first quarter of the 20th century, physicists were not aware of the existence of classical electromagnetic zero-point radiation nor of the importance of special relativity. Inclusion of these aspects allows classical electron theory to be extended beyond its 19th century successes. Here we review spherical electromagnetic radiation modes in a conducting-walled spherical cavity and connect these modes to classical electromagnetic zero-point radiation and to electromagnetic scale invariance. Then we turn to the scattering of radiation in classical electron theory within a simple approximation. We emphasize that, in steady-state, the interaction between matter and radiation is disguised so that the mechanical motion appears to occur without the emission of radiation, even though the particle motion is actually driven by classical electromagnetic radiation. It is pointed out that, for nonrelativistic particles, only the harmonic oscillator potential taken in the low-velocity limit allows a consistent equilibrium with classical electromagnetic zero-point radiation. For relativistic particles, only the Coulomb potential is consistent with electrodynamics. The classical analysis places restrictions on the value of $e^{2}/\hbar c$.

physics.class-ph↗

arXiv2021

Thermal Radiation Equilibrium: (Nonrelativistic) Classical Mechanics versus (Relativistic) Classical Electrodynamics

Energy equipartition is appropriate only for nonrelativistic classical mechanics, but has only limited relevance for a relativistic theory such as classical electrodynamics. In this article, we discuss harmonic-oscillator thermal equilibrium from three different perspectives. First, we contrast the thermal equilibrium of nonrelativistic mechanical oscillators (where point collisions are allowed and frequency is irrelevant) with the equilibrium of relativistic radiation modes (where frequency is crucial). The Rayleigh-Jeans law appears from applying a dipole-radiation approximation to impose the nonrelativistic mechanical equilibrium on the radiation spectrum. In this discussion, we note the possibility of zero-point energy for relativistic radiation, which possibility does not arise for nonrelativistic classical-mechanical systems. Second, we turn to a simple electromagnetic model of a harmonic oscillator and show that the oscillator is fully in radiation equilibrium (which involves all radiation multipoles, dipole, quadrupole, etc.) with classical electromagnetic zero-point radiation, but is not in equilibrium with the Rayleigh-Jeans spectrum. Finally, we discuss the contrast between the flexibility of nonrelativistic mechanics with its arbitrary potential functions allowing separate scalings for length, time, and energy, with the sharply-controlled behavior of relativistic classical electrodynamics with its single scaling connecting together the scales for length, time, and energy. It is emphasized that within classical physics, energy-sharing, velocity-dependent damping is associated with the low-frequency, nonrelativistic part of the Planck thermal radiation spectrum, whereas acceleration-dependent radiation damping is associated with the high-frequency adiabatically-invariant and Lorentz-invariant part of the spectrum corresponding to zero-point radiation.

physics.class-ph↗

arXiv2021

Relativity and Radiation Balance for the Classical Hydrogen Atom in Classical Electromagnetic Zero-Point Radiation

Here we review the understanding of the classical hydrogen atom in classical electromagnetic zero-point radiation, and emphasize the importance of special relativity. The crucial missing ingredient in earlier calculational attempts (both numerical and analytic) is the use of valid approximations to the full relativistic analysis. It is pointed out that the nonrelativistic time Fourier expansion coefficients given by Landau and Lifshitz are in error as the electromagnetic description of a charged particle in a Coulomb potential, and, because of this error, Marshall and Claverie's conclusion regarding the failure of radiation balance is invalid. Rather, using Marshall and Claverie's calculations, but restricted to lowest nonvanishing order in the orbital eccentricity (where the nonrelativistic orbit is a valid approximation to the fully relativistic electromagnetic orbit) radiation balance for classical electromagnetic zero-point radiation is shown to hold at the fundamental frequencies and associated first overtones.

physics.class-ph↗