# Opinions on **Wave–particle duality**

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**Wave–particle duality** is the concept that every elementary particle or quantic entity exhibits the properties of not only particles, but also waves. It addresses the inability of the classical concepts "particle" or "wave" to fully describe the behavior of quantum-scale objects. As Einstein wrote: "*It seems as though we must use sometimes the one theory and sometimes the other, while at times we may use either. We are faced with a new kind of difficulty. We have two contradictory pictures of reality; separately neither of them fully explains the phenomena of light, but together they do*".

Various opinions have arisen about this.

Initiated by Louis de Broglie, before the discovery of quantum mechanics, and developed later as the de Broglie-Bohm theory, the pilot wave interpretation does not regard the duality as paradoxical, seeing both particle and wave aspects as always coexisting. According to Schrödinger the domain of the de Broglie waves is ordinary physical space-time. This formal feature exhibits the pilot wave theory as non-local, which is considered by many physicists to be a grave defect in a theory.

Still in the days of the old quantum theory, another pre-quantum-mechanical version of wave–particle duality was pioneered by William Duane, and developed by others including Alfred Landé. Duane explained diffraction of x-rays by a crystal in terms solely of their particle aspect. The deflection of the trajectory of each diffracted photon was due to quantal translative momentum transfer from the spatially regular structure of the diffracting crystal. Fourier analysis reveals the wave–particle duality as a simply mathematical equivalence, always present, and universal for all quanta. The same reasoning applies for example to diffraction of electrons by a crystal.

In the light of de Broglie's ideas, Erwin Schrödinger developed his wave mechanics by referring the universal wave aspect not to ordinary physical space-time, but rather to a profoundly different and more abstract 'space'. The domain of Schrödinger's wave function is configuration space. Ordinary physical space-time allows more or less direct visualization of cause and effect relations. In contrast, configuration space does not directly display cause and effect linkages. Sometimes, nevertheless, it seemed as if Schrödinger visualized his own waves as referring to ordinary space-time, and there was much debate about this.

Niels Bohr regarded the "duality paradox" as a fundamental or metaphysical fact of nature. Sometimes the wave aspect was apparent, and sometimes the particle aspect, with the same kind of quantic entity, but in respectively different physical settings. He saw it as one aspect of the concept of complementarity. Bohr regarded renunciation of the cause-effect relation, or complementarily, of the space-time picture, as essential to the quantum mechanical account.

Werner Heisenberg considered the question further. He saw the duality as present for all quantic entities, but not quite in the usual quantum mechanical account considered by Bohr. He saw it in what is called second quantization, which generates an entirely new concept of fields which exist in ordinary space-time, causality still being visualizable. Classical field values (e.g. the electric and magnetic field strengths of Maxwell) are replaced by an entirely new kind of field value, as considered in quantum field theory. Turning the reasoning around, ordinary quantum mechanics can be deduced as a specialized consequence of quantum field theory.

Because of the difference of views of Bohr and Heisenberg, the main sources of the so-called Copenhagen interpretation, the position of that interpretation on wave–particle duality is ill-defined.

In a modern perspective, wave functions arise naturally in relativistic quantum field theory in the formulation of free quantum fields. They are necessary for the Lorentz invariance of the theory. Their form and the equations of motion they obey are dictated by under which representation of the Lorentz group they transform.

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Thanks to this graph, we can see the interest Wave–particle duality has and the evolution of its popularity.

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