**6. Conclusion**

field boundaries separated from each other in space-time, while in nonrelativistic

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Because Heisenberg's uncertainty relations use a continuous time parameter, they are only valid when events are defined with respect to specific observers, but not in general for all observers. When interpreted according to Eq. (2) by a fully relativistic theory, we conclude that indeterminacy is due to measurements performed with a non-singular, spatially and temporally extended probe, the photon. This may be compared to the case in classical mechanics of measurements that are performed with a coarsely defined standard. In quantum mechanics, the standard of measurement is the photon, and no matter how high its energy, it cannot be used to localize a point particle more precisely than its wavelength. On the other hand, localizations in atomic clocks occur four-dimensionally with respect to *both*

In the wave mechanical interpretation of quantum mechanics, field boundaries are not specified. Nevertheless fields are a part of wave functions, and field boundaries must be included in a fully relativistic theory of electrodynamics. To satisfy

function composed in this manner as a composite of two physical components may be used to describe the interaction of particles in both bound and free states. Whereas in classical theory forces are three-dimensional vectors with direction and magnitude, in a fully relativistic theory they are four-dimensional and symmetric in the coordinates. They have orientation in space-time, but not direction, with magnitude determined by the instantaneous separation of field boundaries according to (1). Thus force is the continuous application of a discrete form. All interactions of electrodynamics may be conceived of in this way in terms of fields and their

Although the wave function contains all that can be known about a particle, the preceding fully relativistic interpretation of atomic structure indicates the presence of internal characteristics that are *in principle* unknown to observers. The field model, described by (1) and confirmed experimentally by slow or stopped light phenomena, includes internal processes in its description of the wave function that are temporarily restricted from external expression due to field boundaries. The characteristics cannot be accessed because the fields vanish at the field boundaries. Due to the unobserved processes, quantum theory predicts the occurrence of instantaneous action-at-a-distance events such as the collapse of the wave function and other macroscopic phenomena that exist outside of our consciousness. However, if wave functions are interpreted in a fully relativistic theory, we conclude that these phenomena are only unusual when interpreted in abstract space with

The detection events that form the basis of optical theory are due to energy emissions that occur at singular points in time and are referred to as "photons" due to their discrete nature. If energy absorption evolves according to Eq. (2), as the integration of a Lagrangian density over a region of space-time, then excitation is a

!, *t* 

, that are separated by field boundaries. A wave

as combining an electron

quantum mechanics, field boundaries do not exist.

field boundaries, so they occur without measurable error.

that requirement, we interpret the wave function Ψ *r*

!, *t* 

**4.4 Wave mechanics**

boundaries.

**5. Discussion**

**24**

or other particle, and a force *ε r*

respect to continuous time parameters.

It has long been asserted that classical physics is inadequate for describing quantum mechanical phenomena. Consequently experimental results are explained by introducing complementarity and the correspondence principle. However, the problem is not that classical theory is deficient, but it is the insistence on using singularities in a nonrelativistic theory. If the photon's fields are singular, wave and particle properties seem to appear out of nowhere, and experimental results have an intrinsically defined uncertainty. But if the photon is instead described as a localization of fields, uncertainty and duality are accounted for by physical characteristics, fields and field boundaries, and complementarity has a classically derived meaning. A similar explanation is possible for the correspondence principle which specifies the point where a two-particle classical system must be replaced by a three-particle quantum mechanical system to explain what is observed. It may seem to be an acceptable practice to describe particles as singularities propagating and interacting continuously in time, but in a fully relativistic theory the photon cannot be singular. Rather it is a four-dimensional localization of fields defined symmetrically in space-time that determines electron behavior in bound states and also in free space by means of four-dimensional forces.

The mathematical framework surrounding quantum mechanics is precisely the type of description that is expected when a particle of zero mass is absorbed by a two-particle system. The particle properties of the photon are overwhelmed by the other two such that it is impossible to distinguish it independently of them. Sometimes the influence of its continuous properties is more evident (wave mechanics); at other times its discrete properties are prominent (matrix mechanics); and in path integral formulations, the exact field boundaries of bound states are manifested. Each of the three formulations of nonrelativistic quantum mechanics provides a unique perspective to atomic structure by emphasizing a different physical aspect of the three field sources. This may be compared to the simpler three-dimensional practice in architecture of providing three visual perspectives to a building. Each one provides a partial view, and when taken together they give an improved understanding of the structure as a whole. The "whole" of quantum mechanics is given of course by Lagrangian quantum mechanics.

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