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Fundamental Concepts of Quantum Theories

Eliahu Comay

Published Jan 1, 2020• 3 citations

Abstract

The foundations of the mathematical structure of quantum theories of a massive particle are the basis of this analysis.It proves the coherence of the particle-wave duality of quantum theories and the principle of complementarity as well.Furthermore, the noncommutativity of Hermitian operators proves that quantum theories are inherently indeterministic.This feature does not deny the fact that the classical limit of quantum theories agrees with classical physics.It is also shown that the foundations of the mathematical structure of quantum theories impose constraints on any specific quantum theory.It is proved that the first-order Dirac theory is consistent with all constraints.In contrast, second-order theories, such as the Klein-Gordon, the electroweak theory of the W ± and the Z particles, and the Higgs boson theory fail to do that.An analogous analysis proves that also the Majorana neutrino theory is inconsistent with fundamental requirements.Similarly, inconsistencies of Proca's idea about a massive photon are shown.

DOI: 10.4236/oalib.1106970

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The paper examines the Dirac-Pauli differences concerning the order of the primary differential equation of an elementary massive quantum particle.The analysis relies on several self-evident constraints that an acceptable quantum theory must satisfy, like conservation laws, compatibility with Maxwellian electrodynamics and the correspondence principle.The dimension of the quantum function that is used in the Lagrangian density of a given quantum theory together with the corresponding differential equations play an important role in the reasoning procedure.The paper proves that Dirac was right and that second-order quantum theories like the Klein-Gordon equation and the electroweak theory of the W ± bosons do not satisfy fundamental constraints.This outcome is inconsistent with the Standard Model of Particle Physics.

The quantum world has always ignited human curiosity to unveil and speculate on the fascinating secrets of microscopic particles. This curiosity has opened a window to other branches of physics pushing the boundaries of classical understanding. After years of experimentation and analysis, scientists revealed that at microscopic dimensions, Newtonian mechanics failed not only quantitatively but even qualitatively and conceptually, clearing the way for new ideas. The nature of the difference between Newtonian mechanics and quantum mechanics is presented, with the core difference being the inherently probabilistic nature of quantum mechanics. This probabilistic foundation has led to revolutionary concepts like wave-particle duality and entanglement, fundamentally challenging classical ideas of causality and locality. The present article provides a well-focused and comprehensive overview of foundation ideas, concepts, and applications of quantum mechanics that make it principally different from Newtonian mechanics.

Quantum theory comprises a number of physical concepts and methods of calculation applicable to bodies on an atomic scale and consequently to the explanation of the properties of macroscopic bodies in terms of the behaviour of their constituent particles. In advanced work the theory is highly mathematical, but at an elementary level the difficulties lie mostly in the appreciation of the relevant physical ideas. Wave-particle duality, indeterminacy, the nature of a quantized state of a system and the exclusion principle cause most conceptual problems.

This chapter lays out the principles of quantum theory. Quantum theory emerged when experimental results on matter and radiation were found to diverge from the predictions of classical mechanics. It was found that particles had the properties of waves, and that waves had the properties of particles. That duality is taken into account by the Schrödinger equation and the introduction of a wavefunction. One of the great upsets of classical physics is embodied in the uncertainty principle, which expresses the impossibility of the simultaneous specification of certain pairs of properties. One consequence of duality is the extension of microscopy and diffraction to beyond the electromagnetic spectrum.

The paper shows that the variational principle serves as an element of the mathematical structure of a quantum theory. The experimentally confirmed properties of the corpuscular-wave duality of a quantum particle are elements of the analysis. A Lagrangian density that yields the equations of motion of a given quantum theory of a massive particle is analyzed. It is proved that if this Lagrangian density is a Lorentz scalar whose dimension is then the associated action consistently defines the required phase of the quantum particle. The dimension of this Lagrangian density proves that also the quantum function has dimension. This result provides new criteria for the acceptability of quantum theories. An examination of the first order Dirac equation demonstrates that it satisfies the new criteria whereas the second order Klein-Gordon equation fails to do that.