Classical Zero-point Radiation Research Articles

Particle Brownian motion due to random classical radiation: superfluid-like behavior in the presence of classical zero-point radiation

Nonrelativistic classical mechanics allows no fundamental transition between low-velocity and high-velocity forms of behavior, nor between low-temperature and high-temperature forms. In contrast, classical electrodynamics, which is a relativistic theory, allows fundamental transitions in velocity. Furthermore, the inclusion within the classical theory of Lorentz-invariant classical zero-point radiation allows classical electrodynamics to distinguish high-temperature and low-temperature forms of behavior. Because electromagnetism is a relativistic theory, it may provide a thermal radiation bath which gives rise to phenomena in Brownian motion which are not included in a model with a thermal bath based upon nonrelativistic particles. Here we explore the Brownian motion of a classical electric-dipole particle in random classical radiation, making use of the calculations of Einstein and Hopf. The Brownian motion as a function of temperature is analyzed in terms of the mean-square velocity and the diffusion constant for four different classical radiation spectra: the Rayleigh–Jeans spectrum, the Planck spectrum without zero-point radiation, the zero-point radiation spectrum, and the Planck spectrum including zero-point radiation. We illustrate how the inclusion of classical electromagnetic zero-point radiation alters Brownian motion behavior between high-temperature and low-temperature forms. For the Planck spectrum with zero-point radiation, the high-temperature Brownian motion agrees with some aspects found from nonrelativistic mechanics, while the low-temperature behavior includes some aspects analogous to superfluid behavior. At sufficiently low temperatures, the Brownian particle has an increasing mean-square velocity and more rapid diffusion with decreasing temperature due to the increasing dominance of the classical electromagnetic zero-point radiation.

Dirac's classical-quantum analogy for the harmonic oscillator: Classical aspects in thermal radiation including zero-point radiation

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.

Diamagnetic behavior in random classical radiation

Calculations for diamagnetic behavior involving Faraday induction appear in classical electromagnetism textbooks. These calculations give the charged particle motions correctly but then inaccurately introduce the statement that diamagnetism is incompatible with classical thermodynamics, and that quantum theory is required for diamagnetic behavior. Actually, if classical radiative equilibrium in classical zero-point radiation holds before the application of a magnetic field, then it will hold afterwards and will preserve the diamagnetic behavior obtained by the application of Faraday's law. Here, we consider the classical diamagnetism of a charged particle in an isotropic harmonic potential which follows from the four famous spectra of random classical radiation. The zero-point radiation spectrum fully justifies the analysis appearing in the textbooks of classical electromagnetism and in the work of Langevin. The Rayleigh-Jeans spectrum gives no diamagnetic behavior, as is consistent with the Bohr-van Leeuwen theorem. The Planck spectrum without zero-point radiation (surprisingly) gives no magnetic moment at low temperature and paramagnetic behavior at high temperature! Finally, the Planck spectrum with zero-point radiation gives diamagnetic behavior at low temperature and no magnetic moment at high temperature. This last result is in agreement with elementary quantum theory. Once again the Planck spectrum with zero-point radiation provides the best classical description.

Equilibrium for classical zero-point radiation: detailed balance under scattering by a classical charged harmonic oscillator

It has been shown repeatedly over a period of 50 years that the use of relativistic classical physics and the inclusion of classical electromagnetic zero-point radiation leads to the Planck blackbody spectrum for classical radiation equilibrium. However, none of this work involves scattering calculations. In contrast to this work, currently accepted physical theory connects classical physics to only the Rayleigh-Jeans spectrum. Indeed, in the past, it has been shown that a nonlinear classical oscillator (which is necessarily a nonrelativistic scattering system) achieves equilibrium only for the Rayleigh-Jeans spectrum where the random radiation present at the frequency of the second harmonic of the oscillator motion has the same energy per normal mode as the radiation present at the fundamental frequency. Here we continue work emphasizing the importance of relativistic versus nonrelativistic analysis. We consider the scattering of random classical radiation by a charged harmonic oscillator of small but non-zero oscillatory amplitude (which can be considered as a relativistic scattering system) and show that detailed radiation balance holds not only at the fundamental frequency of the oscillator but through the first harmonic corresponding to quadrupole scattering, provided that the radiation energy per normal mode at the first harmonic is double the radiation energy per normal mode at the fundamental frequency. This condition corresponds exactly to the zero-point radiation spectrum which is linear in frequency. It is suggested that for this relativistic scattering system, the detailed balance for zero-point radiation holds not only for the fundamental and first harmonic but extends to all harmonics. Here we have the first example of an explicit relativistic classical scattering calculation; equilibrium corresponds not to the Rayleigh-Jeans spectrum, but rather corresponds to the Lorentz-invariant zero-point radiation spectrum.

Blackbody radiation in classical physics: A historical perspective

We point out that current textbooks of modern physics are a century out-of-date in their treatment of blackbody radiation within classical physics. Relativistic classical electrodynamics including classical electromagnetic zero-point radiation gives the Planck spectrum with zero-point radiation as the blackbody radiation spectrum. In contrast, nonrelativistic mechanics cannot support the idea of zero-point energy; therefore, if nonrelativistic classical statistical mechanics or nonrelativistic mechanical scatterers are invoked for radiation equilibrium, one arrives at only the low-frequency Rayleigh-Jeans part of the spectrum, which involves no zero-point energy, and does not include the high-frequency part of the spectrum involving relativistically invariant classical zero-point radiation. Here, we first discuss the correct understanding of blackbody radiation within relativistic classical physics, and then we review the historical treatment. Finally, we point out how the presence of Lorentz-invariant classical zero-point radiation and the use of relativistic particle interactions transform the previous historical arguments, so as now to give the Planck spectrum including classical zero-point radiation. Within relativistic classical electromagnetic theory, Planck's constant ℏ appears as the scale of source-free zero-point radiation.

Contrasting interactions between dipole oscillators in classical and quantum theories: illustrations of unretarded van der Waals forces

Students encounter harmonic-oscillator models in many aspects of basic physics, within widely varying theoretical contexts. Here we highlight the interconnections and varying points of view. We start with the classical mechanics of masses coupled by springs and trace how the same essential systems are reanalysed in the unretarded van der Waals interactions between dipole oscillators within classical and quantum theories. We note how classical-mechanical ideas from kinetic theory lead to energy equipartition which determines the high-temperature van der Waals forces of atoms and molecules modelled as dipole oscillators. In this case, colliding heat-bath particles can be regarded as providing local hidden variables for the statistical mechanical behaviour of the oscillators. Next we note how relativistic classical electrodynamical ideas conflict with the assumptions of nonrelativistic classical statistical mechanics. Classical electrodynamics which includes classical zero-point radiation leads to van der Waals forces between dipole oscillators, and these classical forces agree at all temperatures with the forces derived from quantum theory. However, the classical theory providing this agreement is not a local theory, but rather a non-local hidden-variables theory. The classical theory can be regarded as involving hidden variables in the random phases of the plane waves spreading throughout space which provide the source-free random radiation.

Classical Zero-Point Radiation and Relativity: The Problem of Atomic Collapse Revisited

The physicists of the early 20th century were unaware of two aspects which are vital to understanding some aspects of modern physics within classical theory. The two aspects are: 1) the presence of classical electromagnetic zero-point radiation, and 2) the importance of special relativity. In classes in modern physics today, the problem of atomic collapse is still mentioned in the historical context of the early 20th century. However, the classical problem of atomic collapse is currently being treated in the presence of classical zero-point radiation where the problem has been transformed. The presence of classical zero-point radiation indeed keeps the electron from falling into the Coulomb potential center. However, the old collapse problem has been replaced by a new problem where the zero-point radiation may give too much energy to the electron so as to cause self-ionization. Special relativity may play a role in understanding this modern variation on the atomic collapse problem, just as relativity has proved crucial for a classical understanding of blackbody radiation.

Contrasting Classical and Quantum Vacuum States in Non-inertial Frames

Classical electron theory with classical electromagnetic zero-point radiation (stochastic electrodynamics) is the classical theory which most closely approximates quantum electrodynamics. Indeed, in inertial frames, there is a general connection between classical field theories with classical zero-point radiation and quantum field theories. However, this connection does not extend to noninertial frames where the time parameter is not a geodesic coordinate. Quantum field theory applies the canonical quantization procedure (depending on the local time coordinate) to a mirror-walled box, and, in general, each non-inertial coordinate frame has its own vacuum state. In complete contrast, the spectrum of random classical zero-point radiation is based upon symmetry principles of relativistic spacetime; in empty space, the correlation functions depend upon only the geodesic separations (and their coordinate derivatives) between the spacetime points. It makes no difference whether a box of classical zero-point radiation is gradually or suddenly set into uniform acceleration; the radiation in the interior retains the same correlation function except for small end-point (Casimir) corrections. Thus in classical theory where zero-point radiation is defined in terms of geodesic separations, there is nothing physically comparable to the quantum distinction between the Minkowski and Rindler vacuum states. It is also noted that relativistic classical systems with internal potential energy must be spatially extended and can not be point systems. Based upon the classical analysis, it is suggested that the claimed heating effects of acceleration through the vacuum may not exist in nature.

The Blackbody Radiation Spectrum Follows from Zero-Point Radiation and the Structure of Relativistic Spacetime in Classical Physics

The analysis of this article is entirely within classical physics. Any attempt to describe nature within classical physics requires the presence of Lorentz-invariant classical electromagnetic zero-point radiation so as to account for the Casimir forces between parallel conducting plates at low temperatures. Furthermore, conformal symmetry carries solutions of Maxwell’s equations into solutions. In an inertial frame, conformal symmetry leaves zero-point radiation invariant and does not connect it to non-zero-temperature; time-dilating conformal transformations carry the Lorentz-invariant zero-point radiation spectrum into zero-point radiation and carry the thermal radiation spectrum at non-zero temperature into thermal radiation at a different non-zero temperature. However, in a non-inertial frame, a time-dilating conformal transformation carries classical zero-point radiation into thermal radiation at a finite non-zero-temperature. By taking the no-acceleration limit, one can obtain the Planck radiation spectrum for blackbody radiation in an inertial frame from the thermal radiation spectrum in an accelerating frame. Here this connection between zero-point radiation and thermal radiation is illustrated for a scalar radiation field in a Rindler frame undergoing relativistic uniform proper acceleration through flat spacetime in two spacetime dimensions. The analysis indicates that the Planck radiation spectrum for thermal radiation follows from zero-point radiation and the structure of relativistic spacetime in classical physics.

Any classical description of nature requires classical electromagnetic zero-point radiation

Any attempt to describe nature within classical physics requires the presence of Lorentz-invariant classical electromagnetic zero-point radiation to account for the Casimir forces between parallel conducting plates at low temperatures. This zero-point radiation leads to classical explanations for several phenomena that are usually regarded as requiring quantum physics. We provide a brief overview of classical electromagnetic theory including classical zero-point radiation and note the areas of agreement and disagreement between the classical and quantum theories, both of which contain Planck’s constant.

Classical physics of thermal scalar radiation in two spacetime dimensions

Thermal scalar radiation in two spacetime dimensions is treated within relativistic classical physics. We first consider an inertial frame in which we give the analogues of Boltzmann’s derivation of the Stefan–Boltzmann law and Wien’s derivation of the displacement theorem using the scaling appropriate to relativistic radiation theory. The spectrum of classical scalar zero-point radiation in an inertial frame is derived both from scale invariance and from Lorentz invariance. We then consider the behavior of thermal radiation in a coordinate frame undergoing (relativistic) constant acceleration. The classical zero-point radiation of inertial frames is transformed to the coordinates of an accelerating frame. Although the two-field correlation function for zero-point radiation at different spatial points at a single time is the same for inertial and accelerating frames, the correlation function at two different times at a single spatial coordinate is different and, in an accelerating frame, has a natural extension to nonzero temperature. The thermal spectrum in the accelerating frame is then transferred back to an inertial frame, giving the familiar Planck result.

Derivation of the Planck spectrum for relativistic classical scalar radiation from thermal equilibrium in an accelerating frame

The Planck spectrum of thermal scalar radiation is derived suggestively within classical physics by the use of an accelerating coordinate frame. The derivation has an analogue in Boltzmann's derivation of the Maxwell velocity distribution for thermal particle velocities by considering the thermal equilibrium of noninteracting particles in a uniform gravitational field. For the case of radiation, the gravitational field is provided by the acceleration of a Rindler frame through Minkowski spacetime. Classical zero-point radiation and relativistic physics enter in an essential way in the derivation which is based upon the behavior of free radiation fields and the assumption that the field correlation functions contain but a single correlation time in thermal equilibrium. The work has connections with the thermal effects of acceleration found in relativistic quantum field theory.

Blackbody Radiation and the Scaling Symmetry of Relativistic Classical Electron Theory with Classical Electromagnetic Zero-Point Radiation

It is pointed out that relativistic classical electron theory with classical electromagnetic zero-point radiation has a scaling symmetry which is suitable for understanding the equilibrium behavior of classical thermal radiation at a spectrum other than the Rayleigh-Jeans spectrum. In relativistic classical electron theory, the masses of the particles are the only scale-giving parameters associated with mechanics while the action-angle variables are scale invariant. The theory thus separates the interaction of the action variables of matter and radiation from the scale-giving parameters. Due to this separation, classical zero-point radiation is invariant under scattering by the charged particles of relativistic classical electron theory. The basic ideas of the matter-radiation interaction are illustrated in a simple relativistic classical electromagnetic example.

Scaling symmetries of scatterers of classical zero-point radiation

Classical radiation equilibrium (the blackbody problem) is investigated by the use of an analogy. Scaling symmetries are noted for systems of classical charged particles moving in circular orbits in central potentials V(r) = −k/rn when the particles are held in uniform circular motion against radiative collapse by a circularly polarized incident plane wave. Only in the case of a Coulomb potential n = 1 with fixed charge e is there a unique scale-invariant spectrum of radiation versus frequency (analogous to zero-point radiation) obtained from the stable scattering arrangement. These results suggest that non-electromagnetic potentials are not appropriate for discussions of classical radiation equilibrium.

Comments on Cole and Zou's Calculation of the Hydrogen Ground State in Classical Physics

Cole and Zou's computer-simulation calculation of the hydrogen ground state in classical electrodynamics with classical electromagnetic zero-point point radiation suggests that the problem of atomic collapse in atomic physics amy have been solved. Analytic calculations of the early 1980s do not contradict the new results.

Classical spinning magnetic dipole in classical electrodynamics with classical electromagnetic zero-point radiation

A classical spinning magnetic dipole is considered within classical electrodynamics with classical electromagnetic zero-point radiation. The stationary probability distribution for the angle of alignment between the spinning magnetic dipole and an external magnetic field is calculated when the system is located in an arbitrary spectrum of random classical radiation. It is found that for the Rayleigh-Jeans spectrum the probability distribution for alignment is just the Boltzmann distribution. However, in classical zero-point radiation the alignment probability distribution is independent of the magnetic field causing alignment. Moreover, for large classical spin angular momentum $Sgg\ensuremath{\hbar}$, the average component of the spin in the direction of alignment is given by $〈{S}_{z}〉=S\ensuremath{-}\frac{1}{2}\ensuremath{\hbar}$, where $\ensuremath{\hbar}$ is Planck's constant (divided by $2\ensuremath{\pi}$) used to set the scale of the classical zero-point radiation spectrum. The results seem suggestive of the idea of space quantization in quantum theory. The model discussed here was first considered by S. Sachidanandam (unpublished).

Derivation of the blackbody radiation spectrum from the equivalence principle in classical physics with classical electromagnetic zero-point radiation

A derivation of Planck's spectrum including zero-point radiation is given within classical physics from recent results involving the thermal effects of acceleration through classical electromagnetic zero-point radiation. A harmonic electric-dipole oscillator undergoing a uniform acceleration $\stackrel{\ensuremath{\rightarrow}}{\mathrm{a}}$ through classical electromagnetic zero-point radiation responds as would the same oscillator in an inertial frame when not in zero-point radiation but in a different spectrum of random classical radiation. Since the equivalence principle tells us that the oscillator supported in a gravitational field $\stackrel{\ensuremath{\rightarrow}}{\mathrm{g}}=\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{\mathrm{a}}$ will respond in the same way, we see that in a gravitational field we can construct a perpetual-motion machine based on this different spectrum unless the different spectrum corresponds to that of thermal equilibrium at a finite temperature. Therefore, assuming the absence of perpetual-motion machines of the first kind in a gravitational field, we conclude that the response of an oscillator accelerating through classical zero-point radiation must be that of a thermal system. This then determines the blackbody radiation spectrum in an inertial frame which turns out to be exactly Planck's spectrum including zero-point radiation.

Derivation of the Planck radiation spectrum as an interpolation formula in classical electrodynamics with classical electromagnetic zero-point radiation

A closely argued derivation of Planck's spectrum for blackbody radiation is presented within classical electrodynamics with classical electromagnetic zero-point radiation. The presence of temperature-independent random classical radiation invalidates the ideas of traditional classical statistical mechanics which become valid only in the low-frequency or high-temperature limit where they lead to the Rayleigh-Jeans law for the thermal spectrum. The assumption of Lorentz invariance for the zero-point radiation determines the high-frequency part of the classical random radiation spectrum. The blackbody problem of classical physics with classical zero-point radiation as considered here is the derivation of an interpolation formula between these high- and low-frequency limits. Here we take advantage of the surprising diamagnetic behavior of a classical free point charge in zero-point radiation. This diamagnetic behavior is compared with the paramagnetic behavior of a magnetic dipole rotor of large moment of inertia, which behavior is derived from the low-frequency form of the radiation spectrum. If one requires the natural simple condition that the diamagnetic behavior as a function of temperature should differ only in the sign of the magnetic moment from the paramagnetic behavior as a function of temperature, then one is led uniquely to the Planck spectrum including zero-point radiation as the equilibrium spectrum for classical random radiation.

Thermal effects of acceleration through random classical radiation

The thermal effects of acceleration found by Davies and Unruh within quantum field theory are shown to exist within random classical radiation. The two-field correlation functions for random classical radiation are used as the basis for investigating the spectrum of radiation observed at an accelerating point detector. An observer with proper acceleration $a$ relative to the Lorentz-invariant spectrum of random classical scalar zero-point radiation finds a spectrum identical with that given by Planck's law for scalar thermal radiation where the temperature is related to the acceleration by $T=\frac{\ensuremath{\hbar}a}{2\ensuremath{\pi}\mathrm{ck}}$. An observer with proper acceleration $a$ relative to the Lorentz-invariant spectrum of random classical electromagnetic radiation finds a stationary radiation spectrum which is not Planck's spectrum. Rather, the observed spectrum in the electromagnetic case contains a term agreeing with Planck's electromagnetic spectrum plus an additional term. This spectrum for the electromagnetic case appears in the work of Candelas and Deutsch for an accelerating mirror and corresponds to thermal radiation in the non-Minkowskian space-time of the accelerating observer. The calculations reported here involve an entirely classical point of view, but are shown to have immediate connections with quantum field theory.

Diamagnetism of a free particle in classical electron theory with classical electromagnetic zero-point radiation

Landau's diamagnetism for a free point charge is shown to exist within classical electron theory with classical electromagnetic zero-point radiation. The system considered is a nonrelativistic classical point charge bound harmonically in three dimensions and situated in a magnetic field when random classical electromagnetic zero-point radiation or thermal radiation is present. The average energy, angular momentum, and magnetic moment are calculated at finite temperature, and then carried to the limit at which the harmonic binding vanishes so as to obtain the behavior for a free particle. In the presence of the Rayleigh-Jeans radiation spectrum one finds that all diamagnetic effects vanish, in agreement with the results of traditional classical statistical mechanics. In the Planck radiation spectrum one finds exactly the results of quantum theory involving a Langevin function for the temperature dependence. In particular, the average angular momentum and magnetic moment for a classical point charge in classical zero-point radiation are $〈{L}_{z}〉=\ensuremath{-}\ensuremath{\hbar}$ and $〈{M}_{z}〉=\ensuremath{-}|e|\frac{\ensuremath{\hbar}}{2\mathrm{mc}}$, where the orientation of the $z$ axis is given by the magnetic field direction. Thus the classical results in the presence of classical zero-point radiation are entirely different from those of traditional classical electron theory, and they suggest possible classical explanations of the space quantization appearing in quantum theory. The results also suggest a derivation of Planck's spectrum from traditional classical statistical mechanics by applying Boltzmann statistics to the orientation of the average magnetic moment caused by the zero-point radiation.