Andrea Valdés Hernández

Disentangling Quantum Entanglement

What is the physical agent that causes noninteracting particles to get entangled? The theory developed in previous chapters is applied in the present one to give due response to this key question. For this purpose, the one-particle treatment presented in Chap. 5 is extendedExtended charge to systems of two particlesExtended particle that are embedded in the common zero-point fieldEntanglement!and zero-point field. The dynamical variables are shown to become correlated when the particles resonate to a common frequency of the background field. When the description is reduced to one in terms of matrices and vectors in the appropriate Hilbert space, the entangled state vectors emerge naturally. For systems of identical particles the properties of invariance of the field variables imply that entanglementEntanglement is maximal and must be described by totally (anti)symmetric states. The results thus obtained are applied to the HeliumHelium atom as a system with two electrons. As a result of entanglement, the total (orbital plus spin) stateHelium!spin states vectors turn out to be antisymmetric. States in which both particles are in the same orbital and spinorial state, are excluded because of the absence of a correlating field mode.

Quantum Mechanics: Some Answers

We are near the end of our journey—though the theory is still far from the finish line, as is only the case with any scientific theory. The time has come, therefore, to recapitulate; recapture the most relevant points of the material presented in previous chapters, summarize some of the answers afforded by present stochastic electrodynamics to the questions posed at the beginning, and put on the table some of the many questions that remain to be explored. The chapter starts therefore with a summary of the most relevant results presented in the body of the book, stressing the genetic power of the zero-point field, on the basis of which the quantum scheme of matter in contact with the radiation field is constructed. Some particular features of quantum mechanics are discussed from this synoptic perspective. A condensed, itemized repertory of the answers offered by present stochastic electrodynamics to the conceptual problems of quantum theory is then provided, followed by a short critical review of the fussy definitions and conceptions of the modern photon. The chapter concludes with some comments on the limitations of the theory here discussed and possibilities for future developments and extensions.

Quantum Mechanics: Some Questions

This initial chapter is devoted to a brief, critical review of the major conceptual difficulties that permeate the whole of quantum mechanics, especially when it is interpreted within the framework of the (mainstream or orthodox) Copenhagen school. The text is written for a reader who is interested in these issues and ready to accept that no interpretationInterpretation of quantum mechanics is free of conceptual difficulties, which require some repair. A short overview of the contents of the book is included at the end of the chapter, guided by the leitmotif of the theory presented, namely, that quantization can be understood as an emergent phenomenon arising from a deeper stochastic process. Specifically, the permanent interaction between matter and the zero-point radiation field is shown, chapter by chapter, to give rise to quantum features of both, field and matter. An appendix to the chapter provides a concise introduction to the Probability interpretation!ensembleensemble Ensembleinterpretation ofProbability interpretation probability, a much extended Probabilities!extendednotion among physicists, but hardly discussed in the literature.

The Road to Heisenberg Quantum Mechanics

This chapter takes us through an alternative itinerary to quantum mechanics, on this occasion to the Heisenberg formulation of the theory. Our starting point is again the stochastic Abraham-Lorentz equationAbraham-Lorentz!equation for the particle embedded in the zero-point field. A detailed analysis of the stationary solutions of this equation exhibits the resonant response Linear sed!resonant responseof the particle to certain modes of the field, determined in each case by the problem itself. The condition of ergodicity Matrix mechanics!and ergodicity Energy-balance condition!and ergodicity of these stationary states has far-reaching consequences, both for the physical behavior of the system and for the formalism used to describe it. In particular, the response of the particle to the field turns out to be linear, regardless of the nonlinearities of the external force. Further, the dynamical variables become represented by matrices, which satisfy the Linear sed!Heisenberg equationsHeisenberg equation. The statistical nature of the description becomes evident. The ensuing fundamental commutator \([x,p]=i\hbar \) represents a direct measure of the intensity of the Commutator!and correlationfluctuations impressed by the zero-point field upon the particle. The path followed in the derivations throws thus new light on the physical meaning of the Linear sed!Heisenberg descriptionquantities involved in the Heisenberg description.Harmonic oscillator!Heisenberg description

Causality, Nonlocality, and Entanglement in Quantum Mechanics

This chapter takes us into the domains of quantum nonlocality. The journey starts with a brief introduction to Bohm’s causal theory of quantum mechanics, which serves to further discuss the nonlocality contained in the Schrodinger descriptionQuantum regime!and Schrodinger description. A critical discussion of this causal and deterministic approach illustrates the virtues and limitations associated with the BohmianCausality interpretationInterpretation of quantum mechanics. The tools developed in previous chapters are then applied to a more detailed analysis of quantum nonlocality, both for single-particle and bipartite systems. Some important results are derived, which throw light on the relationship between the quantum potentialQuantum potential and linearity, fluctuations, and nonlocalityQuantum potential!and nonlocality. The bipartite system is further analyzed and the connections between nonlocality, entanglementEntanglement and noncommutativity are disclosed for general continuous variables.

The Long Journey to the Schrödinger Equation

This chapter marks the beginning of an itinerary that will take us to the quantum theory of matter. All along this exciting journey we will be accompanied by the zero-point radiation field introduced in Chap. 3, considered as a real, fluctuating field in permanent interaction with matter. Our point of departure is the (nonrelativistic) Abraham-Lorentz!equationAbraham-Lorentz equation governing the particle motion under the action of this field plus an arbitrary external (binding, conservative) force. A statistical treatment leads to a generalized phase-space Fokker-Planck equation!and laws of evolutionFokker-Planck equation. In the transition to configuration space through a partial averaging, a hierarchy of equations is obtained for the local moments of the momentum. When a balance is eventually reached in the mean between the energy lost by the particle through radiation reaction Radiation reactionand the energy gained by it from the background field, any remaining effect of the radiation terms becomes negligible. In this (time-asymptotic) limit, theFokker-Planck equation!generalized generalized Fokker-Planck equation!and laws of evolutionFokker-Planck equation transforms into a true Fokker-Planck equationFokker-Planck equation, describing a Markov processMarkov process. Further, in the Approximation!radiationlessradiationless approximationRadiationless approximation Linear sed!radiationless approximation the first two equations of the hierarchy decouple from the rest and are shown to be equivalent to Schrodinger’s equation. The main lessons and implications of these results are discussed. In particular, an explanation is given for the impossibility to go back from the Schrodinger description Quantum regime!and Schrodinger descriptionand retrieve a fullDetailed energy balance!and Schrodinger equation phase-space description that is consistent with quantum mechanics.

The Planck DistributionPlanck distribution, a Necessary Consequence of the Fluctuating Zero-Point Field

The historical problem of the spectral distributionDistribution of the radiation field in equilibrium at a given temperature is revisited in this chapter. Taking into account the inescapable presence of the fluctuatingFluctuations!zero-point zero-point radiation field, a derivation of Planck’s formula is presented that does not introduce any quantum postulate. The starting point is Wien’sWien!law law, which dictates that the mean energy of a Harmonic oscillator!and spinharmonic oscillator of frequency \(\omega \)—or rather, the mean energy of a monochromatic mode of the radiation field—must be proportional to the frequency. The description obtained through a standard thermodynamic analysis is shown to be incomplete, as it does not provide for the existence of fluctuations at zero temperature. This limitation is lifted by means of a statistical analysis that allows to determine the variance of the Fluctuationsfluctuations of the energy at \(T=0.\) A combination of the outcomes of the two analyses leads to a differential equation, which upon integration gives the Planck distribution Planck distributionat any temperature. The different terms contributing Commutator!and correlationto the energy fluctuations acquire thus a new meaning, different from those assigned to it by Planck and by EinsteinEinstein, A.. The implications of the results regarding the question of continuity or discontinuity of the field energy are discussed, and the chapter concludes with a brief discussion on the reality Realityof the zero-point fluctuations and the origin of quantum fluctuations.

Beyond the Schrödinger Equation

In this chapter we exploit the possibilities of stochastic electrodynamics to carry the theory beyond the usual limits of Schrodinger’s quantum mechanics and enter the province of (nonrelativistic) quantum electrodynamics. The equation of evolution for the mean energy, derived in Chap. 4, is employed to obtain the formulas for the radiative lifetimesLifetime and make contact with the Einstein A and B coefficients.Zeropoint!and A-B coefficients A-B coefficients Further, small correctionsQuantum corrections to the energy of the atomic levels are shown to be due to the fluctuationsFluctuations of the dynamical variables. Without the need to use perturbation theory—because the zero-point fieldFluctuations!zero-point field is there from the beginning—the correct formulas are obtained for the Lamb shiftLamb shift Cavity effects!on the Lamb shiftand other radiative effects, to lowest order. The second part of the chapter is devoted to an inquiry about the origin of the spinSpin of the electronPolarization!and spin. Just as it gives rise to position, momentumMomentum and energyZeropoint!energy fluctuationsZeropoint!energy fluctuations Spin!and fluctuations Fluctuations!momentum, the zero-point field is seen to induce an angular momentum resulting from the instantaneous torque exerted by the Lorentz forceLorentz!force on the particle. A close analysis based on the separation of the field modes of given circular polarization Polarization!circularreveals the existence of a spin angular momentum of value \(\hbar /2\), and of a corresponding magnetic momentMoment!magnetic Magnetic moment with a \(g\)-factor of value 2. This allows us to identify the spin of the electron as a further emergent property, generated by the interaction of the particle with the zero-point field.

The Phenomenological Stochastic Approach: A Short Route to Quantum Mechanics

In preparation for the chapters that follow, this one offers an introduction to the study of quantum mechanics as a stochastic process, in the form of a basic phenomenological theory knownNelson theory as stochastic (or Nelson) mechanics, or Stochastic quantum mechanicsstochastic quantum mechanics. The kinematics of the process is developed first, using a description of the motion of particles that involves finite time intervals \(\Delta t\) and up to second-order derivatives; this kinematics is made in terms of two different velocities and four accelerations. An extended ‘Newton’s Second Law’ involving a linear combination of these accelerations, leads under appropriate conditions to two equations, one being equivalent to the Continuity equationcontinuity equation and the second one representing a general dynamical law. The Schrodinger equationDetailed energy balance!and Schrodinger equation follows from this set of equations for a particular selection of the parameters. By ignoringCetto, A. M. the source of the stochasticity, this theory does not provide the elements to derive the value of the parameters; this is shown to be one of its main limitations. It has, however, the value of providing an intuitive physical image of the stochastic process underlying quantum mechanics. Moreover, by showing that the Brownian process implies an altogether different choice of the parameters, it establishes a neat distinction between classical and quantum stochastic processesQuantum stochastic process.

The Zero-Point Field Waves (and) Matter

Before our journey comes to an end, we propose to look at one more trace of the zero-point field on matter. In previous chapters, most important aspects of the behavior of quantum systems have been shown to carry the signature of this field. However, one feature that has been left aside so far is the de BroglieDe Broglie, L. wave, just the creature that gave life to the quantum description of matter. De Broglie’s wavelength is reconstructed from the perspective of the present theory and shown to be associated with a wave of electromagnetic nature. This physical notion of theDe Broglie’s wave de Broglie wave is applied to the simplest example of all, the infinite square well, in an effort to refine our image of the motion of particles taking place in this system. Further, the ‘matter diffraction’ by a double slit is explained by arguing that what is really diffracted is the background field, its diverted rays ‘guiding’ the particles to form the well-known diffraction pattern on the far-away screen. Finally, on the other extreme of the scale, a cosmological model is put forward for the zero-point field. It is shown that this permanent component of the background radiation can be considered the result of a regenerative process, sustained by the radiation of all the charges in the Universe; this model explains thus a well-known ‘strange’ relation between the large numbers.