Semiclassical Description Research Articles

Unifying Monopole and Center Vortex as the Semiclassical Confinement Mechanism.

Magnetic excitations play a crucial role in understanding the color confinement of 4D Yang-Mills theory, and we have the monopole and the center vortex as plausible candidates to explain its mechanism. Under suitable compactified setups of 4D Yang-Mills theory, we can achieve different weakly coupled descriptions of confinement phenomena: The monopole mechanism takes place on R^{3}×S^{1} with the double-trace deformation, and the center-vortex mechanism is effective on R^{2}×T^{2} with the 't Hooft flux. We unify these two semiclassical descriptions by showing the explicit relation between the monopole and center vortex.

Low and moderate x gluon contribution to exclusive Compton scattering processes

Abstract We revisit the high energy semi-classical description of the exclusive processes DVCS, TCS, and Double DVCS by explicitly keeping track of the Feynman x dependence in both the hard and the hadronic matrix elements. This is achieved by a modification of the standard shock wave approximation to derive the effective Feynman rules, which leads to a generic expression on which we then perform a partial twist expansion to get rid of quantities suppressed by the proper physical scales. We obtain a compact factorized master formula that can be used to investigate the Bjorken limit at leading twist. In particular, we recover the full one-loop result in the collinear limit for pure gluon exchange with the target. Finally, we discuss the subtleties in taking the simultaneous collinear and small x limit.

Complementarity for a dynamical black hole

Black hole complementarity posits that the interior of a black hole is not independent from its Hawking radiation. This leads to an apparent violation of causality: the interior can be acausally affected by operators acting solely on the radiation. We argue that this perspective is misleading and that the black hole interior must be viewed as existing in the causal past of the Hawking radiation, despite the fact that they are spacelike separated in the semiclassical description. Consequently, no operation on the Hawking radiation—no matter how complex—can affect the experience of an infalling observer. The black hole interior and the radiation only appear spacelike separated in the semiclassical description because an infalling observer’s ability to access complex information is limited; the chaotic dynamics on the horizon, as viewed from the exterior, then converts any effect caused by such an observer to information in the Hawking radiation which cannot be accessed at the semiclassical level. We arrive at the picture described above by considering a unitary exterior description in which the flow of information is strictly causal, which we extend to apply throughout the entire history of black hole evolution, including its formation. This description uses the stretched event horizon as an inner edge of spacetime, on which the information inside is holographically encoded. We argue that the global spacetime picture arises from coarse graining over black hole microstates, and discuss its relationship with the exterior description. Published by the American Physical Society 2024

Aharonov-Bohm effect mediated by massive photons

Virtual photons play an essential role in the locally realistic description of the Aharonov-Bohm interference. We show that the effect of virtual photons in the interferometer is manifested by a change in their spectrum. In particular, when a vacuum is confined between two ideal conducting plates, the photons obey the two-dimensional Proca equation, the wave equation with finite effective mass. This results in a short-range interaction between a test charge and a magnetic flux, and hence the Aharonov-Bohm effect is reduced exponentially at a large distance between the two bodies. On the other hand, a semiclassical description is also possible, and this raises the interesting question of how to prove the physical reality of virtual photons.

Nonequilibrium Transition between Dissipative Time Crystals

We show a dissipative phase transition in a driven nonlinear quantum oscillator in which a discrete time-translation symmetry is spontaneously broken in two different ways. The corresponding regimes display either discrete or incommensurate time-crystal order, which we analyze numerically and analytically beyond the classical limit, addressing observable dynamics, phenomenology in different (laboratory and rotating) frames, Liouvillian spectral features, and quantum fluctuations. Via an effective semiclassical description, we show that phase diffusion dominates in the incommensurate time crystal (or continuous time crystal in the rotating frame), which manifests as a band of eigenmodes with a lifetime growing linearly with the mean-field excitation number. Instead, in the discrete time-crystal phase, the leading fluctuation process corresponds to quantum activation with a single mode that has an exponentially growing lifetime. Interestingly, the transition between these two regimes manifests itself already in the quantum regime as a spectral singularity, namely, as an exceptional point mediating between phase diffusion and quantum activation. Finally, we discuss this transition between different time-crystal orders in the context of synchronization phenomena. Published by the American Physical Society 2024

Semiclassics for the QCD vacuum structure through T2-compactification with the baryon-’t Hooft flux

We study QCD vacuum structure with the topological θ angle using a recently proposed semiclassical approach on ℝ2 × T2 with the ’t Hooft and baryon magnetic fluxes. Under the assumption of adiabatic continuity in this setup, the confining vacuum can be described by the dilute gas of center vortices. With this semiclassical approach, we derive the 2d effective description at small T2 and successfully explain the reasonable theta dependence of the QCD vacuum: in the one-flavor QCD at θ = π, the CP symmetry is spontaneously broken for quark mass above a critical value and restored for a subcritical mass, while the CP symmetry is always spontaneously broken in the multi-flavor QCD at θ = π. From our semiclassical description, we discuss implications to the 4d chiral Lagrangian and propose how the η′ meson should be incorporated in consistent with known global structures: the periodicity of the η′ should be extended from the naive one 2π to 2πN. Additionally, we revisit the phase diagram of Nf = 1 + 1 and Nf = 1 + 1 + 1 QCD on the up and down quark mass plane, confirming and refining the existence of the CP-broken Dashen phase.

Complexity growth and the Krylov-Wigner function

For any state in a D-dimensional Hilbert space with a choice of basis, one can define a discrete version of the Wigner function — a quasi-probability distribution which represents the state on a discrete phase space. The Wigner function can, in general, take on negative values, and the amount of negativity in the Wigner function has an operational meaning as a resource for quantum computation. In this note, we study the growth of Wigner negativity for a generic initial state under time evolution with chaotic Hamiltonians. We introduce the Krylov-Wigner function, i.e., the Wigner function defined with respect to the Krylov basis (with appropriate phases), and show that this choice of basis minimizes the early time growth of Wigner negativity in the large D limit. We take this as evidence that the Krylov basis (with appropriate phases) is ideally suited for a dual, semi-classical description of chaotic quantum dynamics at large D. We also numerically study the time evolution of the Krylov-Wigner function and its negativity in random matrix theory for an initial pure state. We observe that the negativity broadly shows three phases: it rises gradually for a time of OD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ O\\left(\\sqrt{D}\\right) $$\\end{document}, then hits a sharp ramp and finally saturates close to its upper bound of D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\sqrt{D} $$\\end{document}.

Holographic reflected entropy and islands in interface CFTs

We investigate the reflected entropy for various mixed state configurations in the two dimensional holographic conformal field theories sharing a common interface (ICFTs). In the AdS3/ICFT2 framework, we compute the holographic reflected entropy for the required configurations in the vacuum state of the ICFT2 which is given by twice the entanglement wedge cross section (EWCS) in a spacetime involving two AdS3 geometries glued along a thin interface brane. Subsequently, we evaluate the EWCS in the bulk geometry involving eternal BTZ black strings with an AdS2 interface brane, which is dual to an ICFT2 in the thermofield double (TFD) state. We explore the system from a doubly holographic perspective and determine the island contributions to the reflected entropy in the two dimensional semi-classical description involving two CFT2s coupled to an AdS2 brane. We demonstrate that the results from the island formula match precisely with the bulk AdS3 results in the large tension limit of the interface brane. We illustrate that the phase structure of the reflected entropy is quite rich involving many novel induced island phases and demonstrate that it obeys the expected Page curve for the reflected entropy in a radiation bath coupled to the AdS2 black hole.

Microscopic Origin of the Entropy of Astrophysical Black Holes.

We construct an infinite family of microstates for black holes in Minkowski spacetime which have effective semiclassical descriptions in terms of collapsing dust shells in the black hole interior. Quantum mechanical wormholes cause these states to have exponentially small, but universal, overlaps. We show that these overlaps imply that the microstates span a Hilbert space of log dimension equal to the event horizon area divided by four times the Newton constant, explaining the statistical origin of the Bekenstein-Hawking entropy.

Single field inflation in the light of Pulsar Timing Array Data: quintessential interpretation of blue tilted tensor spectrum through Non-Bunch Davies initial condition

In this work, we present a quintessential interpretation of having a blue-tilted tensor power spectrum for canonical single-field slow-roll inflation to explain the recently observed Pulsar Timing Array (NANOGrav 15-year and EPTA) signal of Gravitational Waves (GW). We formulate the complete semi-classical description of cosmological perturbation theory in terms of scalar and tensor modes using the Non-Bunch Davies initial condition. We found that the existence of the blue tilt (nt)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(n_t)$$\\end{document} within the favoured range 1.2<nt<2.5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1.2<n_t<2.5$$\\end{document} can be explained in terms of a newly derived consistency relation. Further, we compute a new field excursion formula using the Non-Bunch Davies initial condition, that validates the requirement of Effective Field Theory in the sub-Planckian regime, |Δϕ|≪Mpl\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|\\Delta \\phi |\\ll M_{\ extrm{pl}}$$\\end{document} for the allowed value of the tensor-to-scalar ratio, r<0.06\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r<0.06$$\\end{document} from CMB observations. In our study, we refer to this result as Anti Lyth bound as it violates the well-known Lyth bound originally derived for Bunch Davies initial condition. Further, we study the behaviour of the spectral density of GW and the associated abundance with the frequency, which shows that within the frequency domain 10-9Hz<f<10-7Hz\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$10^{-9}~{\ extrm{Hz}}<f<10^{-7}~{\ extrm{Hz}}$$\\end{document} the outcome obtained from our analysis is completely consistent with the Pulsar Timing Array (NANOGrav 15-year and EPTA) signal. Also, we found that the behaviour of GW spectra satisfies the CMB constraints at the low frequency, f∗∼7.7×10-17Hz\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f_*\\sim 7.7\ imes 10^{-17}~{\ extrm{Hz}}$$\\end{document} corresponding to the pivots scale wave number, k∗∼0.05Mpc-1.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k_*\\sim 0.05~{\ extrm{Mpc}}^{-1}.$$\\end{document} Finally, the sharp falling behaviour of the GW spectra within the frequency domain 10-7Hz<f<1Hz\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$10^{-7}~{\ extrm{Hz}}<f<1~{\ extrm{Hz}}$$\\end{document} validates our theory in the comparatively high-frequency regime as well.

Flux correlators and semiclassics

We consider correlators for the flux of energy and charge in the background of operators with large global U(1) charge in conformal field theory (CFT). It has recently been shown that the corresponding Euclidean correlators generically admit a semiclassical description in terms of the effective field theory (EFT) for a conformal superfluid. We adapt the semiclassical description to Lorentzian observables and compute the leading large charge behavior of the flux correlators in general U(1) symmetric CFTs. We discuss the regime of validity of the large charge EFT for these Lorentzian observables and the subtleties in extending the EFT approach to subleading corrections. We also consider the Wilson-Fisher fixed point in d = 4 − ϵ dimensions, which offers a specific weakly coupled realization of the general setup, where the subleading corrections can be systematically computed without relying on an EFT.

A principle of maximum ignorance for semiclassical gravity

The principle of maximum ignorance posits that the coarse-grained description of a system is maximally agnostic about its underlying microscopic structure. We briefly review this principle for random matrix theory and for the eigenstate thermalization hypothesis. We then apply this principle in holography to construct ensembles of random mixed states. This leads to an ensemble of microstates which models our microscopic ignorance, and which on average reproduces the effective semiclassical physics of a given bulk state. We call this ensemble the state-averaging ansatz. The output of our model is a prediction for semiclassical contributions to variances and higher statistical moments over the ensemble of microstates. The statistical moments provide coarse-grained — yet gravitationally non-perturbative — information about the microstructure of the individual states of the ensemble. We show that these contributions exactly match the on-shell action of known wormhole configurations of the gravitational path integral. These results strengthen the view that wormholes simply parametrize the ignorance of the microstructure of a fundamental state, given a fixed semiclassical bulk description.

Effects of a generalized uncertainty principle on the MIT bag model equation of state

AbstractThe Generalized Uncertainty Principle (GUP) is motivated by the premise that spacetime fluctuations near the Planck scale impose a lower bound on the achievable resolution of distances, leading to a minimum length. Inspired by a semiclassical method that integrates the GUP into the partition function by deforming its phase space, we induce a modification on the thermodynamic quantities of the MIT bag model that we propose serves as an effective semiclassical description of deconfined quark matter in a space with minimal length. We investigate the consequences of this deformation on the zero‐temperature limit, revealing a saturation limit for the energy density, pressure, and baryon number density and an overall decrease of the thermodynamic quantities which suggests an enhanced stability against gravitational collapse. These findings extend existing research on GUP‐deformed Fermi gases. Ultimately, our description introduces the effects of quantum gravity in the equations of state for compact stars in a mathematically simple manner, suggesting the potential for extension to more complex systems.

Unified light-matter Floquet theory and its application to quantum communication

Periodically driven quantum systems can exhibit a plethora of intriguing nonequilibrium phenomena that can be analyzed using Floquet theory. Naturally, Floquet theory is employed to describe the dynamics of atoms interacting with intense laser fields. However, this semiclassical analysis cannot account for quantum-optical phenomena that rely on the quantized nature of light. In this paper, we take a significant step to go beyond the semiclassical description of atom-photon coupled systems by unifying Floquet theory with quantum optics using the framework of full-counting statistics. This is achieved by introducing counting fields that keep track of the photonic dynamics. This formalism, which we dub “photon-resolved Floquet theory” (PRFT), is based on two-point tomographic measurements, instead of the two-point projective measurements used in standard full-counting statistics. Strikingly, the PRFT predicts the generation of macroscopic light-matter entanglement when atoms interact with multimode electromagnetic fields, thereby leading to complete decoherence of the atomic subsystem in the basis of the Floquet states. This decoherence occurs rapidly in the optical-frequency regime, but is negligible in the radio-frequency regime. Our results thus pave the way for the design of efficient quantum memories and quantum operations. Finally, employing the PRFT, we propose a quantum-communication protocol that can significantly outperform the state-of-art few-photon protocols by two orders of magnitude or better. The PRFT potentially leads to insights in various Floquet settings including spectroscopy, thermodynamics, quantum metrology, and quantum simulations. Published by the American Physical Society 2024

Cosmology from random entanglement

We construct entangled microstates of a pair of holographic CFTs whose dual semiclassical description includes big bang-big crunch AdS cosmologies in spaces without boundaries. The cosmology is supported by inhomogeneous heavy matter and it partially purifies the bulk entanglement of two disconnected auxiliary AdS spacetimes. We show that the island formula for the fine grained entropy of one of the CFTs follows from a standard gravitational replica trick calculation. In generic settings, the cosmology is contained in the entanglement wedge of one of the two CFTs. We then investigate properties of the cosmology-to-boundary encoding map, and in particular, its non-isometric character. Restricting our attention to a specific class of states on the cosmology, we provide an explicit, and state-dependent, boundary representation of operators acting on the cosmology. Finally, under genericity assumptions, we argue for a non-isometric to approximately-isometric transition of the cosmology-to-boundary map for “simple” states on the cosmology as a function of the bulk entanglement, with tensor network toy models of our setup as a guide.

Zombie cats on the quantum-classical frontier: Wigner-Moyal and semiclassical limit dynamics of quantum coherence in molecules.

In this paper, we investigate the time evolution of quantum coherence-the off-diagonal elements of the density matrix of a multistate quantum system-from the perspective of the Wigner-Moyal formalism. This approach provides an exact phase space representation of quantum mechanics. We consider the coherent evolution of nuclear wavepackets in a molecule with two electronic states. For harmonic potentials, the problem is analytically soluble for both a fully quantum mechanical description and a semiclassical description. We highlight the serious deficiencies of the semiclassical treatment of coherence for general systems and illustrate how even qualitative accuracy requires higher order terms in the Moyal expansion to be included. The model provides an experimentally relevant example of a molecular Schrödinger's cat state. The alive and dead cats of the exact two-state quantum evolution collapse into a "zombie" cat in the semiclassical limit-an averaged behavior, neither alive nor dead, leading to significant errors. The inclusion of the Moyal correction restores a faithful simultaneously alive and dead representation of the cat that is experimentally observable.

Numerically Exact Solution for a Real Polaritonic System under Vibrational Strong Coupling in Thermodynamic Equilibrium: Loss of Light-Matter Entanglement and Enhanced Fluctuations.

The first numerically exact simulation of a full ab initio molecular quantum system (HD+) under strong ro-vibrational coupling to a quantized optical cavity mode in thermal equilibrium is presented. Theoretical challenges in describing strongly coupled systems of mixed quantum statistics (bosons and Fermions) are discussed and circumvented by the specific choice of our molecular system. Our numerically exact simulations highlight the absence of zero temperature for the strongly coupled matter and light subsystems, due to cavity-induced noncanonical conditions. Furthermore, we explore the temperature dependency of light-matter quantum entanglement, which emerges for the ground state but is quickly lost already in the deep cryogenic regime. This is in contrast to predictions from the Jaynes-Cummings model, which is the standard starting point to model collective strong-coupling chemistry phenomenologically. Moreover, we find that the fluctuations of matter remain modified by the quantum nature of the thermal and vacuum-field fluctuations for significant temperatures, e.g., at ambient conditions. These observations (loss of entanglement and coupling to quantum fluctuations) have implications for the understanding and control of polaritonic chemistry and materials science, since a semiclassical theoretical description of light-matter interaction becomes reasonable, but the typical (classical) canonical equilibrium assumption for the nuclear subsystem remains violated. This opens the door for quantum fluctuation-induced stochastic resonance phenomena under vibrational strong coupling, which have been suggested as a plausible theoretical mechanism to explain the experimentally observed resonance phenomena in the absence of periodic driving that has not yet been fully understood.

Standing spin waves in finite quantum spin spiral chains

Standing spin waves in quantum spin-spiral chains are investigated using the local Pn and total absorbed power P. This method allows the investigation of the spin wave pattern and dispersion even if the quantum mechanical spin system initially has no magnetic order due to quantum fluctuations. In addition, the spin dynamics during resonance were examined more closely. Without taking damping into account, a multi-spin Rabi oscillation can be observed. The Rabi Oscillation disappears when damping is taken into account. The cause of this behavior becomes apparent when looking at the spin wave excitation in the semi-classical description. In the semi-classical description, also a Rabi oscillation occurs, which will be prevented by damping. The cause of the semi-classical Rabi oscillation is to be found in the effective field, which periodically drives the spins out of the equilibrium position or brings them in. The damping creates a spin torque directed against the effective field’s spin torque which results in small spin oscillations around its equilibrium position.

AdS3 orbifolds, BTZ black holes, and holography

Conical defects of the form (AdS3 × S3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathbbm{S}}^3 $$\\end{document})/ℤk have an exact orbifold description in worldsheet string theory, which we derive from their known presentation as gauged Wess-Zumino-Witten models. The configuration of strings and fivebranes sourcing this geometry is well-understood, as is the correspondence to states/operators in the dual CFT2. One can analytically continue the construction to Euclidean AdS3 (i.e. the hyperbolic ball ℍ3+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathbb{H}}_3^{+} $$\\end{document}) and consider the orbifold by any infinite discrete (Kleinian) group generated by a set of elliptic elements γi ∈ SL(2, ℂ), γiki\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\gamma}_i^{{\ extrm{k}}_i} $$\\end{document} = \U0001d7d9, i = 1, . , K. The resulting geometry consists of multiple conical defects traveling along geodesics in ℍ3+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathbb{H}}_3^{+} $$\\end{document}, and provides a semiclassical bulk description of correlation functions in the dual CFT involving the corresponding defect operators, which is nonperturbatively exact in α′. The Lorentzian continuation of these geometries describes a collection of defects colliding to make a BTZ black hole. We comment on a recent proposal to use such correlators to prepare a basis of black hole microstates, and elaborate on a picture of black hole formation and evaporation in terms of the underlying brane dynamics in the bulk.

The zitterbewegung electron puzzle

This work is an updated revision of semiclassical descriptions for the electron, including the fully relativistic QED-P model from H. J. Wilson based on the original Dirac equation (DE). The models presented hereafter go beyond the depiction of the electron as a structureless nondimensional point like charge with momentum and position determined by a probabilistic interpretation of the wavefunction described in terms of an electronic density cloud. These models share features in common that provide useful insights concerning the nature of the electron; for instance, they all consider zitterbewegung, a light speed “trembling-along-the-way” electron motion, to be a real oscillatory motion of the electron. The last model presented in this review is the electron mass model from Val Baker et al. [Phys. Essays 32, 255 (2019)], where the electron mass is defined in terms of a holographic surface-to-volume ratio Φ and the relationship of the electric charge at the Planck scale to that at the electron scale, obtaining a value in agreement with the latest CODATA value. We discuss the relationship between these models. The large number of correspondences between the models should not be taken lightly and indicate, in our view, that something very fundamental about the nature of the electron is being put forward by this study.