The paper shows that the use of mathematical imaginaries, complex numbers, and even more so quaternions in the Quantum Theory to increase the dimension of solutions of one-dimensional equations is, in principle, incorrect. As a result, Einstein's condition was violated: SOME equations of Classical Physics can be rewritten in operator form. The use of correct multidimensional models makes it possible, using the example of the Harmonic Oscillator, to obtain the Planck-Einstein Quantization for da Broglie waves.
The paper shows that the use of mathematical imaginaries, complex numbers, and even more so quaternions in the Quantum Theory to increase the dimension of solutions of one-dimensional equations is, in principle, incorrect. As a result, Einstein's condition was violated: SOME equations of Classical Physics can be rewritten in operator form. The use of correct multidimensional models makes it possible, using the example of the Harmonic Oscillator, to obtain the Planck-Einstein Quantization for da Broglie waves.
This chapter shows that the use of mathematical imaginaries, complex numbers, and even more so quaternions in quantum theory to increase the dimension of solutions of one-dimensional equations is, in principle, incorrect. As a result, Einstein’s condition was violated: SOME equations of classical physics can be rewritten in operator form. The use of correct multidimensional models makes it possible, using the example of the harmonic oscillator, to obtain the Planck-Einstein quantization for da Broglie waves.
The correspondence between classical and quaternion quantum equations, permits considering the universe (vacuum) as an ideal elastic solid. Elementary particles would have to be standing or soliton-like waves. Tension induced by the compression and twisting of the elastic medium would increase the energy density, consequently generate a gravity forcing and affect the wave speed. Consequently the gravity could be described by an index of refraction. Theory was created by combining the Cauchy model of the elastic continuum with the Planck-Kleinert crystal hypothesis. The quaternion-imaginary Lagrange’an, the quaternion motion equation and the quaternionic oscillator allowed deriving: - The Schrödinger equation from the functional integral, which identifies the quaternion-imaginary quantum Hamiltonian. - The 2nd order wave equation system describing both the bosons and the gravity in terms of quaternionic Poisson equation. - The first order quaternionic wave equation system. - The family of the second order wave equation systems describing both the particles and the generated quaternionic force-fields (four-potentials). - The fundamental Planck and gravity constants. - The quaternionic continuity equation in an ideal elastic solid.
Ideal resonance does not fade, and thus, it is, as it were, out of time. Therefore, the corresponding Planck QUANTUM, which determines the STATIONARY allowed states of energy, is simply a reflection of the ideal resonance. So the corresponding Planck QUANTUM, which determines the STATIONARY Allowed States of Energy, is simply a reflection of the Ideal Resonance. And modern “Quantum Mechanics,” which moved away from both Planck’s Quantization and the correct use of the Mechanical Model, simply used a simplified analysis of the Mechanical Harmonic Oscillator. The fundamental difference of QUANTUMS is only that for frequencies not equal to zero and their minimum energy of the resonance-quantum has a finite, non-zero initial value, which also determines the minimum distance between the allowed energy levels. At the same time, the internal structure of the QUANTUM can not only be electromagnetic, like Planck’s, and not only be acoustic, like Einstein’s, but also be determined by the so-called de Broglie matter waves. However, in order to remove any mysticism from QUANTUMS, one must UNDERSTAND that the DUALISM introduced by them is determined by Pontryagin’s functional connection of the ELEMENTARY PARTICLE and the packet of waves that form it.
Ideal Resonance does not fade, and thus, it is, as it were, out of time. So the corresponding Planck QUANTUM, which determines the STATIONARY Allowed States of Energy, is simply a reflection of the Ideal Resonance. which "Quantum Mechanics" used on the example of the Mechanical Harmonic Oscillator. And the fundamental difference of QUANTUMS is only that for frequencies not equal to zero and their minimum energy of the Resonance-Quantum has a finite, non-zero initial value, which also determines the minimum distance between the allowed energy levels. At the same time, the internal structure of the QUANTUM can be not only electromagnetic, like Planck's, and not only acoustic, like Einstein's, but also be determined by the so-called de Broglie matter waves. But, in order to remove any mysticism from QUANTUMS, one must UNDERSTAND that the DUALISM introduced by them is determined by Pontryagin's functional connection of the ELEMENTARY PARTICLE and the Packet of Waves that form it.
Abstract Although the classical examples of harmonic oscillators such as springs or the simple pendulum are of limited use in the submicroscopic world of quantum mechanics, there are plenty of other applications that make the oscillator worth studying. Atoms in a crystal, for instance, oscillate about their mean positions and it is their vibrations that carry sound and heat through the material. Indeed, we know from classical theory that by using normal coordinates we may regard any system performing small oscillations about a point of stable equilibrium as a collection of harmonic oscillators. This extends even to the normal modes of vibration of a string or of the electromagnetic field. The harmonic oscillator thus provides the key to understanding how wave phenomena such as light can be quantized, and lies at the heart of Einstein’s idea of the photon and of Planck’s quantum.
In principle, the lattice of elementary propositions of a generic quantum system admits a representation in real, complex or quaternionic Hilbert spaces as established by Solèr’s theorem (1995) closing a long standing problem that can be traced back to von Neumann’s mathematical formulation of quantum mechanics. However up to now there are no examples of quantum systems described in Hilbert spaces whose scalar field is different from the set of complex numbers. We show that elementary relativistic systems cannot be described by irreducible strongly-continuous unitary representations of SL(2, C) on real or quaternionic Hilbert spaces as a consequence of some peculiarity of the generators related with the theory of polar decomposition of operators. Indeed such a ”naive” attempt leads necessarily to an equivalent formulation on a complex Hilbert space. Although this conclusion seems to give a definitive answer to the real/quaternionic-quantum-mechanics issue, it lacks consistency since it does not derive from more general physical hypotheses as the complex one does. Trying a more solid approach, in both situations we end up with three possibilities: an equivalent description in terms of a Wigner unitary representation in a real, complex or quaternionic Hilbert space. At this point the ”naive” result turns out to be a definitely important technical lemma, for it forbids the two extreme possibilities. \nIn conclusion, the real/quaternionic theory is actually complex. This improved approach is based upon the concept of von Neumann algebra of observables. Unfortunately, while \nthere exists a thorough literature about these algebras on real and complex Hilbert spaces, an analysis on the notion of von Neumann algebra over a quaternionic Hilbert space is \ncompletely absent to our knowledge. There are several issues in trying to define such a mathematical object, first of all the inability to construct linear combination of operators \nwith quaternionic coeffients. Restricting ourselves to unital real *-algebras of operators we are able to prove the von Neumann Double Commutant Theorem also on quaternionc \nHilbert spaces. Clearly, this property turns out to be crucial.