Luis de la Peña

Statistical consequences of the zero-point energy of the harmonic oscillator

In a recent thermodynamic analysis of the harmonic oscillator Boyer has shown, using an interpolation procedure, that the existence of a zero-point energy leads to Planck’s law. We avoid the interpolation procedure by adding a statistical argument to arrive at Planck’s law as a consequence of the existence of the zero-point energy. As in Boyer’s argument, no explicit assumption of quantum mechanics is introduced. We discuss the relation of our results to the analysis of Planck and Einstein which led to the notion of the quantized radiation field. We then inquire into the discrete or continuous behavior of the energy and pinpoint the origin and meaning of the discontinuities. To include zero-point fluctuations (which are neglected in the thermodynamic analysis), we discuss the statistical (in contrast to the purely thermodynamic) description of the oscillator, which accounts for both the thermal and temperature-independent contributions to the dispersion of the energy.

The free particle as described by stochastic electrodynamics

Abstract Stochastic electrodynamics applied to an extended electron reproduces the basic properties of the quantum-mechanical free particle.

Linear stochastic electrodynamics: Looking for the physics behind quantum theory

In this chapter, which covers part of the course given at ELAF, a straight-forward procedure is presented that leads from the basic postulates of stochastic electrodynamics to the usual formalism of quantum theory. The theory thus developed is called linear stochastic electrodynamics, to underline that one of its basic features is the (asymptotic) linear response of atomic systems to the background field. The chapter starts with a brief discussion of some open questions in quantum theory and of the possibility to find an answer to them by resorting to the zeropoint radiation field as the source of the quantum behavior of matter. The basic properties of this field are discussed, and a brief enumeration is made of some of the positive results and vital shortcomings of standard stochastic electrodynamics. After identifying the source of these shortcomings in the assumption that the background field is not altered by its interaction with matter, linear stochastic electrodynamics is developed and shown to lead...

The Wave Properties of Matter

With the theory thus far developed it has been possible to address two of the most conspicuous properties of the quantum domain, namely, the random behaviour of matter and the quantization phenomenon, both of them efficiently and succinctly (though quite cryptically) expressed by the Heisenberg equations of motion augmented with the quantum rule. However, there is a third and most remarkable quantum feature that has been practically ignored in all our previous considerations, namely, the undulatory behaviour of matter. Of course, with formal manipulations of the results obtained so far it would be possible to ascertain the wave content of quantum mechanics; but this would appear to reduce it to a mere accident due to mathematical coincidences, and it would add little if anything to our physical understanding of the related phenomena. This is clearly an unsatisfactory situation, considering the fundamental nature that the wave properties of quantum systems are considered to have.

Genesis of quantization of matter and radiation field

Are we to accept quantization as a fundamental property of nature, the origin of which does not require or admit further investigation? To get an insight into this question we consider atomic systems as open systems, since they are by necessity in contact with the electromagnetic radiation field. This includes not only photonic radiation, but, more importantly for our purposes, the random zero-point or nonthermal radiation that pervades the Universe. The Heisenberg inequalities, atomic stability and the existence of discrete solutions are explained as a result of the permanent action of this field upon matter and the balance between mean absorbed and emitted powers in the equilibrium regime. A detailed study carried out along the years has led to the usual quantum-mechanical formalism as a powerful and revealing statistical description of the behavior of matter in the radiationless approximation, as well as to the radiative corrections of nonrelativistic QED. The theory presented gives thus a response to the question posed above, within a local, realist and objective framework: quantization appears as an emergent phenomenon due to the matter-field interaction.

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.