High-Temperature Plasma in Casimir Physics

Abstract

We present a brief review of a nontraditional but significant application for a high-temperature charged plasma. The unorthodox proposition was made by Barry Ninham concerning a contribution from Casimir forces across high-temperature electron–positron plasma in nuclear interactions. The key message in this review is that high temperatures, about ten to the eleventh kelvin, are found to be essential. Certainly, classical, semi-classical, and quantum considerations for the background media impact both the Casimir effect and the physics of stars and the Universe.

Keywords: Lifshitz forces; Casimir effect; lifetime of mesons; electron–positron plasma

1. Introduction

In the years following Casimir’s findings, quantum vacuum fluctuation-induced forces have been investigated thoroughly, both theoretically and experimentally. Vacuum and thermal fluctuations of the quantized field around a molecule or between surfaces differ from free space, resulting in attractive or repulsive intermolecular interactions.

A valid concept of the nature of molecular forces was first put forward in 1894 by Peter (Pyotr) Lebedev. The following is stated about Lebedev’s consideration: “In Hertz’s researches, in his interpretation of light oscillations as electromagnetic processes, there lies another problem which has hitherto not been considered, the problem of the sources of radiation, of the processes which take place in a molecular vibrator when it radiates light energy into space.” Meanwhile, Lebedev stated the following: “This problem takes us, on the one hand, into the field of spectral analysis and, on the other, quite unexpectedly, into the theory of molecular forces, one of the most complicated problems of modern physics. This follows from the following considerations. From the standpoint of the electromagnetic theory of light, it must be admitted that between two light-emitting molecules, as between two vibrators in which electromagnetic oscillations arise, there exist mechanical forces, caused by the electrodynamic interaction of the alternating electric currents in the molecules (according to Ampere’s law), or of the alternating charges in them (according to Coulomb’s law). We must therefore admit that in this case there exist intermolecular forces whose origin is closely connected with radiation processes.”

Surface force measurements and theoretical advancement of the Lifshitz formula of interaction energy, resulting from vacuum fluctuation of a quantized field, expanded to encompass magnetic and conductive particles, as well as liquids between dissimilar surfaces. In the 1970s, Barry Ninham and collaborators at the Australian National University initiated these developments. The development and practical implementation of the theory of intermolecular forces, and its experimental verification, even reportedly led three of the main contributors to be short-listed for a Nobel Prize in chemistry. Actually, recently, a considerable amount of both novel and less novel studies have been conducted on modeling Casimir, Lifshitz, and van der Waals forces. A thorough discussion on this was presented by Bo Sernelius. Theories of intermolecular dispersion forces have been investigated so extensively that relatively little remains to be addressed. However, even lately, new applications are arising.

Here, we review how the temperatures relevant for stellar physics may, indeed, be in the same range as those predicted for Casimir–Yukawa forces between a pair of neutrons in an atomic nucleus. A recent series of papers published by Ninham and his colleagues proposed an impact from high-temperature plasma on the Casimir forces across intervening electron–positron plasma during nuclear interactions. Certainly, classical, semi-classical, and quantum considerations for the background media may impact the Casimir effect at the nuclear scale and in the physics of stars and the Universe.

2. High-Temperature Electron–Positron Plasma and Casimir–Yukawa Forces

2.1. Can Meson Physics Be Linked to Casimir Theory?

In this section, we address Casimir forces between particles in the presence of background plasma. We focus on the predicted temperatures relevant for the fundamental quantum electrodynamics of nuclear interactions. In our model we assume that the interactions among nuclear particles occur inside plasma composed of fluctuating, continuously created and destroyed electron–positron pairs. Nuclear particle interactions are generally described as having a screened Yukawa potential. Ninham and the author of this paper employed an approximation to compare the screened Casimir potential with the absolute asymptotic expression of this potential across the electron–positron plasma between surfaces. Significant similarities were observed that suggested a potential contribution of screened quantum vacuum interaction between the surfaces to the interactions between the nuclear particles. Ninham and collaborators proposed estimating two nucleons as a pair of reflecting spheres approximated by two perfectly conducting plates with nucleon cross-sectional areas. The zero-temperature Casimir interaction energy and force are given by equation (1): the energy is minus pi squared over seven hundred and twenty times the cube of the separation, times the reduced Planck constant and the speed of light, and the force is minus pi squared over two hundred and forty times the fourth power of the separation, times the same constants.

Here the separation is the surface distance between two protons. The estimated surface area is pi times the square of the proton radius, taken to be 0.8 femtometres. The surface-to-surface distance between two nucleons is estimated to be one fermi. The binding energy between two nucleons, resulting from vacuum fluctuations of the quantized field, is around 5 million electronvolts. The binding energies per nucleon typically range from 1.1 million electronvolts to 8.8 million electronvolts. This result was already known in the early 1970s and discussed in an unpublished manuscript by Ninham and Colin Pask, recently accepted for publication in a historical journal.

The problem is somewhat similar in spirit to some ideas by Hendrik Casimir for the stability of charged electrons. Repulsive forces arise between several surface areas due to the distribution of negative charges on the surface of the electron. An attractive force has to balance this repulsive force in order to keep the electron stable and give it a finite size. Casimir proposed that zero-point energy from vacuum fluctuations of a quantized field may generate the attractive force known as Poincaré stress. Inspired by this consideration, various calculations of Casimir energy have been reported, all of which conclude that the magnitude of the interaction is right but with the incorrect sign. It gives a further repulsive force.

Ninham and the author of this paper demonstrated almost two decades ago that screened Casimir interactions may contribute to nuclear interactions. Interestingly, when a non-relativistic plasma is investigated, the relativistic energy — the particle mass times the speed of light squared — appears in the interaction energy in a curious way: it replaces the temperature. In a high-temperature system of about ten to the eleventh kelvin, the strength of interactions or fluctuations often depends on Boltzmann’s constant times the temperature. In the plasma model, the same dependence arises, but with the rest energy appearing in the same place, as if the energy scale controlling the interaction were set by the particle’s rest energy rather than the thermal energy. This shows that intriguing physics might not be apparent in the issue, emphasizing the importance of including a relativistic mass from the start.

2.2. The Casimir Interaction Energy Between Perfectly Reflecting Surfaces Across a Charged Plasma

As proposed by Lebedev, the origin of the Casimir interaction between metal surfaces, via fluctuations in the electromagnetic modes, is closely connected with radiation processes. The full formalism of quantum electrodynamics, including the formidable theory for intermolecular forces from Evgeny Lifshitz and collaborators, is rather complicated. The impact of the semi-classical theory is largely due to the feature that much of the quantum electrodynamics formalism can be obtained using Maxwell’s equations by imposing proper boundary conditions when attributing to each quantized electromagnetic mode its zero-temperature ground-state energy and the free energy at finite temperature. This point was pioneered, and explored, by Ninham and V. Adrian Parsegian, with colleagues.

For two identical planar objects in a medium, the reflection coefficients for transverse magnetic and transverse electric modes for a wave incident from the first medium onto the interface with the second are equations (2) and (3): the transverse magnetic coefficient is the difference of the cross products of the two dielectric functions with the two propagation factors, over their sum; the transverse electric coefficient is the difference of the two propagation factors over their sum. Those propagation factors are defined by equation (4) as the square root of the squared transverse wavevector minus the dielectric function times the squared ratio of frequency to the speed of light.

The Casimir–Lifshitz energy between two identical planar surfaces interacting through a medium is equation (5): the reduced Planck constant times a sum over the two polarisations, of an integral over the transverse wavevector and over frequency, of the logarithm of one minus the exponential of minus twice the propagation factor times the separation, times the squared reflection coefficient at imaginary frequency. This is the internal energy at zero temperature. The interaction energy at finite temperature is equation (6), the same expression with Boltzmann’s constant times the temperature replacing the reduced Planck constant and the frequency integral replaced by a sum over discrete Matsubara frequencies, where the prime denotes that the zero-order term must be halved. The Matsubara frequencies are given by equation (7).

Ninham and Pask observed that the Casimir interaction at zero temperature, arising from vacuum fluctuation of the quantized field, is sufficient to provide the binding energy of nucleons within a nucleus. Ninham and collaborators then considered the effects of temperature, equation (8), which reduces the free energy to a single integral over the wavevector of the logarithm of one minus an exponential in the combined wavevector and Matsubara frequency.

Using that expression, in the absence of plasma between two planar surfaces, Ninham and John Daicic explicitly derived the interaction energy, equation (9): minus pi squared times the reduced Planck constant times the speed of light over seven hundred and twenty times the cube of the separation, minus a term in the Euler–Riemann zeta function of three times the cube of Boltzmann’s constant times the temperature, plus a term in pi squared times the separation times the fourth power of Boltzmann’s constant times the temperature, and so on.

These are derived for the case of two perfectly conducting reflecting planes, with a transverse electric reflection coefficient of minus one and a transverse magnetic coefficient of one, with a vacuum — that is, an absence of plasma — between the plates. The expression describes the high-temperature limit of the free energy in the case where no plasma is present between the planes. It is worth noting that the first term represents the attractive Casimir energy at zero temperature. The third term represents the black body radiation energy, in a vacuum and at equilibrium, between the plates. This result contradicts the attractive Casimir interaction energy term reported in the earlier work. Following that previous study we assume that the first and third terms are equal at equilibrium. From this one finds the temperature based on the separation distance between the plates: the temperature equals the reduced Planck constant times the speed of light, divided by twice Boltzmann’s constant and the separation, where the attractive force balances out repulsive forces. Notably, distances of one to two femtometres correspond to a temperature range of about ten to the eleventh to ten to the twelfth kelvin. Curiously, this is in the same temperature range as predicted, with a completely different theory and entirely different underlying physics, for critical temperatures for Bose filled star interiors.

As previously identified in this section, the second term in equation (9) is the chemical potential term associated with the Gibbs free energy. This can be identified based on the existence of electron–positron pair plasma, formed by the photon-mediated process in which an electron and a positron convert to and from a photon in the gap. Using the temperature at a given separation distance, one can obtain the electron–positron pair density in the plasma. As discussed in the monograph, the numbers of electrons and positrons in this plasma are nearly equal and both numbers are exceptionally large even at temperatures of the order of the rest energy. At higher densities, the electron–positron plasma behaves rather like an ideal gas, allowing ideal gas equations to be used while ignoring interparticle interactions. The second term in equation (9) can be analyzed further using the given density of electron–positron plasma, equation (10), which equates the chemical potential term to pi times the total pair density times the reduced Planck constant times the speed of light, over six, where the total density is the sum of the electron and positron densities. This way of expressing the chemical potential term leads to the equivalence which is looked for.

For two perfectly reflecting planes, the vacuum fluctuation energy across a dissipation-free plasma reads equation (11), obtained by inserting the dielectric function for an electron–positron plasma — one plus the squared plasma frequency over the squared Matsubara frequency — into the propagation factor. The zero-frequency term in the Matsubara sum takes the alternative form of equation (12), an integral from the inverse plasma screening length upwards, where that screening parameter is the plasma frequency divided by the speed of light.

The derivation of an asymptotic Casimir interaction across a plasma was a nontrivial task carried out more than twenty-five years ago by Ninham, and much later published as an appendix. At fixed separation distance for sufficiently high temperatures, or at fixed temperatures other than absolute zero and for sufficiently large separations, one finds the expansion of equation (13), whose leading term is minus Boltzmann’s constant times the temperature times the screening parameter times an exponential of minus twice that parameter times the separation, over four pi, with a correction term, followed by a term quadratic in the temperature carrying two further exponentials, and corrections of higher exponential order.

The vacuum fluctuation energy terms for both the zero-frequency and the higher-frequency parts exhibit behavior similar to that of the Yukawa potential. Both terms contribute to the Casimir–Yukawa binding energy, remarkably aligning with the experimentally reported binding energy per nucleon. An extension that took into account the magnetic permeability of an electron–positron pair was recently explored by the authors and colleagues.

Due to electromagnetic fluctuation interactions, the Casimir–Yukawa binding energy at this separation comprises a contribution of minus 0.9 million electronvolts coming from the zero-frequency term, and minus 3.6 million electronvolts coming from the higher-frequency terms, therefore producing a total binding energy of 4.5 million electronvolts. As the separation between nucleons decreases, the binding energy increases, reflecting the variation in binding energies among different nuclei. This behavior aligns with the influences of the local environment on the internal structure of nucleons. The binding energy per nucleon changes between atomic nuclei, ranging from 1.1 million electronvolts in deuterium to 8.8 million electronvolts in nickel-62.

2.3. The Klein–Gordon Equation and Semi-Classical Estimates for the Meson Mass

The vacuum fluctuation interaction energy associated with perfectly reflecting plates within plasma can be calculated using Maxwell’s equations, which, after a Fourier transform and exploiting the expression for the dielectric function for an electron–positron plasma, reduces to equation (14): the Laplacian of the potential plus the squared ratio of frequency to the speed of light, times one minus the squared ratio of plasma frequency to frequency, times the potential, equals zero.

Hideki Yukawa proposed that the interaction between nuclear particles could be obtained from the Klein–Gordon equation, which has as its solution a potential proportional to the squared gauge coupling constant of the meson and fermion fields, times an exponential decaying with a scaling constant related to the mass of the exchanged meson, divided by the separation. The Yukawa potential is, after a Fourier transformation, given by equation (15), which has the same form as equation (14) with the squared ratio of the pion rest energy to the reduced Planck constant in place of the squared plasma frequency.

Gian Carlo Wick proposed that mesons operate through the emission and absorption of virtual excitations, with the time taken for the excitation to traverse between a pair of nucleons being measured from the range of the nuclear force divided by the speed of light. The relativistic energy, being at least the pion rest energy, which adheres to the Heisenberg uncertainty principle for energy and time, gives the range as the reduced Planck constant divided by the pion mass times the speed of light. From equations (14) and (15), one identifies the plasma frequency with the pion rest energy divided by the reduced Planck constant. From this, one can find the meson mass as equation (16): twice the electron charge times the reduced Planck constant, divided by the speed of light, times the square root of pi times the electron density over the electron rest energy.

From that expression, one finds the plasma frequency in terms of the electron density and mass in the standard way. Then the meson mass has been estimated to be 267 electron masses, which is in agreement with the experimentally reported value of 264 electron masses.

2.4. Lifetime of Plasmons and Mesons

In the consideration in Section 2.2 above, we assumed that the zero-point vacuum energy and the black body radiation energy completely cancel each other at equilibrium. The remaining entities are collective excitations, specifically plasmons inside the residual electron–positron plasma. These plasmons were recognized as neutral pions. Subsequently, we estimated the lifetime of a semi-classical analogue to the neutral pion. In this lifetime, a plasmon decays into two electron–positron pairs. These can undergo decay to provide two photons. The expansion of the plasmon peak and its duration, which is at least the reciprocal of that width, are both theoretically established and empirically assessed, and are given by equation (17), an expression in the Fermi energy, the ratio of the plasmon wavevector to the Fermi wavevector, and the ratio of the plasmon energy to twice the Fermi energy.

In that expression the Fermi energy scales as the two-thirds power of the density, the plasma frequency as the square root of the density, and the Fermi wavevector as the cube root of the density. All three have explicit dependence on density, and in our current study, these quantities have been shown to be dependent on the separation distance between the nucleons. According to Ninham, it is possible to relate the wave vector with the electron and positron densities without further approximations. However, after failing to find such a relation, to obtain the lifetime of the plasmon, Ninham and colleagues employed an estimate inspired by Wick’s arguments discussed briefly in Section 2.3 above. Specifically, the reasoning to correlate the wavevector with energy was employed. The relativistic energy associated with plasmon excitation, that of a meson of the pion mass, is assumed to be partitioned into the kinetic energy of each particle in two electron–positron pairs. This results in an approximation for the plasmon wave vector: at most the speed of light times the square root of half the product of the pion and electron masses, divided by the reduced Planck constant.

The estimate yielded an identical numerical value, to the first decimal place, as the naive — Weinberg’s term — quantum field theory approximation for the uncharged pion lifetime. Both our findings for the lifetime and the naive estimate exhibit the same order of magnitude, about 0.2 times ten to the minus sixteen seconds. This can be compared with the quantum field theory result of about 0.80 to 0.852 times ten to the minus sixteen seconds, which matches the experimentally obtained result of about 0.834 times ten to the minus sixteen seconds.

3. Future Outlooks

We observed that advancing this field requires the extension of these concepts of nuclear interactions to incorporate a relativistic plasma response function and magnetic, or spin, susceptibilities. If the arguments supporting the contribution of vacuum fluctuation interactions to nuclear and meson physics — which partially relate nuclear and electromagnetic interactions — are considered valid, it suggests that decomposition of nuclear forces into Coulomb and nuclear contributions may require revision. The issue appears to hold equal significance to that encountered in physical chemistry. The established theories are predicated on the assumption that electrostatic forces, analyzed via a nonlinear framework, and electrodynamic forces, examined using the linear approximation of Lifshitz theory, are distinct and separate. The ansatz contravenes both the Gibbs adsorption equation and the gauge requirement pertaining to the electromagnetic field. In future research, a model assumption to be considered is that the charged negative and positive pions emerge as bound states of electron with plasmon and positron with plasmon.

4. Discussion: Critical Temperatures in Casimir–Yukawa Theory

In this mini-review, we discuss predicted critical temperatures based on the nucleon–nucleon Casimir–Yukawa contribution to nuclear binding energy, lifetime, and meson mass. Remarkably, Casimir forces act between protons and neutrons on the nuclear scale. Notably, the temperatures related to the creation of an electron–positron plasma in Casimir–Yukawa semi-classical theory for nuclear interactions are found to be of the same order of magnitude as the estimated critical temperatures for the creation of a charged Bose–Einstein stellar core. We observed that this temperature impacts meson mass and the Casimir–Yukawa potential contribution to the nuclear binding energy. In the considered theories, it has to be stressed that density, energy, and temperature are closely linked. This is similar to the appealing observation by Wick that meson mass is related to the relevant distances via an uncertainty principle.

5. Final Remarks

The current understanding of nuclear and particle physics apparently indicates — for instance, via detailed first-principle lattice quantum chromodynamics simulations — that the essential features of nuclear forces arise from the quark and gluon degrees of freedom described by quantum chromodynamics. The long-range behavior of the nuclear force may be consistent with the pion-exchange potential. Since lattice quantum chromodynamics simulations often neglect quantum electrodynamic interactions, it has generally been assumed that electrons, positrons, and photons are not required to describe the main features of nuclear forces at the energy scale of pion exchange. Therefore, the model first presented by Ninham and one of the authors of this review has been considered somewhat speculative. However, there is certainly enough energy from Casimir forces available to account for nucleon interactions. And, if it does not give a contribution to the canonical theory, where has that energy gone? There is more to this point: it is known that nuclear particles, both protons and neutrons, are polarizable particles. It has even been shown that this polarizability can impact trajectories of nuclear particles in the vicinity of other nuclear particles.

To conclude, it is of historical interest that applications of the electrodynamical Casimir effect within nuclear physics have been proposed in the past. One can, for instance, recall the MIT quark bag model, which shows that zero-point fluctuations for gluons and quarks could generate zero-point energy for quarks. This serves as a phenomenological framework for fitting experimental data. A recognized review in this field is that of Peter Hasenfratz and Julius Kuti. Similar work was carried out by Iver Brevik in 1986. There are always roads to discovery for those who dare to challenge known and established research.

Author Contributions, Funding and Acknowledgements

All authors contributed equally to this mini-review article. All authors have read and agreed to the published version of the manuscript.

This research is part of project number 2022/47/P/ST3/01236 co-funded by the National Science Centre and the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement number 945339. Institutional and infrastructural support for the ENSEMBLE3 Centre of Excellence was provided through the ENSEMBLE3 project, delivered within the Foundation for Polish Science International Research Agenda Programme and co-financed by the European Regional Development Fund and the Horizon 2020 Teaming for Excellence initiative, as well as the Polish Ministry of Education and Science initiative “Support for Centres of Excellence in Poland under Horizon 2020”.

There were no numerical data used or generated in the current article. We dedicate this work to Barry W. Ninham ahead of his 90th birthday. The authors declare no conflicts of interest.

(The fifty-item reference list is omitted; the complete text is at the source.)