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Paper# EmQM13: Emergence of quantization: the spin of the electron

Ana María Cetto, Universidad Nacional Autónoma de México, MX#### Abstract

In the course of the later years our small group has concentrated on a problem that is simple to state but complex to solve. It can be synthesized in the question: How much of the quantum world emerges by adding to classical physics the random zero-point radiation field (zpf)? To arrive at an answer as complete and convincing as possible we have approached the question via different routes, which complement and reinforce each other.

Firstly we study the problem of the continuous radiation field in equilibrium with matter at a fixed temperature, taking into account that the field includes the zero-point component. This from the outset violates classical physics, since the energy equipartition among the field oscillators does not any more apply. The thermo-statistical study of the problem leads, without the need to include any postulate of quantization or discontinuity, to the Planck distribution for the equilibrium field. This means that the field has been quantized, in the usual language.

Secondly we study what happens to submicroscopic matter in permanent interaction with the zpf. For this purpose we reduce the problem to its mechanical part by constructing the Fokker-Planck equation in the phase space of the (radiating) particle. The entire analysis is carried out in the nonrelativistic, electric dipole approximation. In the time-asymptotic limit, when the system has reached a state of equilibrium, a reduced statistical description in configuration space is obtained, which under the condition of energy balance becomes exactly the Schrödinger description, in the radiationless approximation. An alternative route focuses on the properties of the stationary solutions of the stochastic Abraham-Lorentz equation, which under the demand of ergodicity exhibit a linear, resonant response of the system to certain field modes. The treatment in terms of these stationary solutions leads naturally to the Heisenberg formalism, the said resonance frequencies corresponding to the frequencies of transition between stationary states.

The Fokker-Planck equation serves also to derive a set of evolution equations for the average values of important dynamical variables, which generalize Ehrenfest’s theorem. Explicit formulas are thus obtained in the asymptotic (i.e., quantum) regime for the radiative corrections (to lowest order in the fine structure constant), giving correct predictions in all cases. Further, the fluctuations impressed by the field on the mechanical subsystem are shown to engender the spin of the electron, as a result of the interaction of the particle with circularly polarized modes of the zero-point field. The theory is local from the outset; yet in its reduced description it contains as a natural term the so-called quantum potential, which is responsible for contributions having a nonlocal appearance and usually interpreted as manifestations of a real nonlocality. Analogously, in the case of systems containing two particles there appears an effective interaction mediated by specific modes of the zpf; this ‘nonclassical’ interaction leads to the entanglement of particles. When the reference to the zpf is neglected, as happens systematically in usual quantum theory, such interaction appears as nonlocal.

In summary, the zero-point field turns out to be an essential ingredient for the emergence of quantization, and due consideration of it helps explain a number of weird quantum features in causal, local, objective and realistic terms. Thus the answer to our question above opens up interesting and important questions for further research.

This work has received financial support from DGAPA-UNAM through project PAPIIT IN106412-2.