New Symmetries, Conserved Quantities and Gauge Nature of a Free Dirac Field

We present and amplify some of our previous statements on non-canonical interrelations between the solutions to free Dirac equation (DE) and Klein–Gordon equation (KGE). We demonstrate that all the solutions to the DE (possessing point- or string-like singularities) can be obtained via differentiation of a corresponding pair of the KGE solutions for a doublet of scalar fields. In this way, we obtain a “spinor analogue” of the mesonic Yukawa potential and previously unknown chains of solutions to DE and KGE, as well as an exceptional solution to the KGE and DE with a finite value of the field charge (“localized” de Broglie wave). The pair of scalar “potentials” is defined up to a gauge transformation under which corresponding solution of the DE remains invariant. Under transformations of Lorentz group, canonical spinor transformations form only a subclass of a more general class of transformations of the solutions to DE upon which the generating scalar potentials undergo transformations of internal symmetry intermixing their components. Under continuous turn by one complete revolution the transforming solutions, as a rule, return back to their initial values (“spinor two-valuedness” is absent). With an arbitrary solution of the DE, one can associate, apart from the standard one, a non-canonical set of conserved quantities, positive definite “energy” density among them, and with any KGE solution-positive definite “probability density”, etc. Finally, we discuss a generalization of the proposed procedure to the case when the external electromagnetic field is present.

Decoding the wave particle duality

The action reaction principle (ARP), a fundamental ingredient of Physics, is taken for granted, because it is automatically fulfilled along the ordinary Hamiltonian, classical or quantum, time evolution law. But in quantum mechanics, there is an extraordinary evolution law, the projection of state rule along quantum measurements, which is not Hamiltonian. Consequently, the ARP is not automatically fulfilled along quantum measurements, and it must be checked case by case. Surprisingly, very simple quantum measurements, both theoretical processes and experiments, show an apparent violation of the ARP, so that the hidden reaction must be found. In the analyzed experiment, the ARP is restored if some new system, the quantum or de Broglie wave, exists and locally interacts with the detector. There cannot be interaction at a spatial distance between the particle (photon or electron) and the obstacle‐detector.

Relativistic electrodynamics without Lorentz transformations

The object of this exercise is to show that the force between moving charges can be obtained in a very different way than is usual without recourse to the Lorentz transformations. We suppose the spinning electron creates two massless strings which connect to another electron either stationary or moving. Each string carries a wave, one the de Broglie wave and the other a wave that moves at c that mediates the force between charges in addition to guiding the electron’s motion.

A review of de Broglie particle–wave mechanical systems

The existence of the so-called ‘dark’ issues of mechanics implies that our present accounting for mass and energy is incorrect in terms of applicability on a cosmological scale, and the question arises as to where the difficulty might lie. The phenomenon of quantum entanglement indicates that systems of particles exist that individually display certain characteristics, while collectively the same characteristic is absent simply because it has cancelled out between individual particles. It may therefore be necessary to develop theoretical frameworks in which long-held conservation beliefs do not necessarily always apply. The present paper summarises the formulation described in earlier papers (Hill, JM. On the formal origin of dark energy. Z Angew Math Phys 2018; 69:133-145; Hill, JM. Some further comments on special relativity and dark energy. Z Angew Math Phys 2019; 70: 5–14; Hill, JM. Special relativity, de Broglie waves, dark energy and quantum mechanics. Z Angew Math Phys 2019; 70: 131–153.), which provides a framework that allows exceptions to the law that matter cannot be created or destroyed. In these papers, it is proposed that dark energy arises from conventional mechanical theory, neglecting the work done in the direction of time and consequently neglecting the de Broglie wave energy [Formula: see text]. These papers develop expressions for the de Broglie wave energy [Formula: see text] by making a distinction between particle energy [Formula: see text] and the total work done by the particle [Formula: see text], that which accumulates from both a spatial physical force [Formula: see text] and a force [Formula: see text] in the direction of time. In any experiment, either particles or de Broglie waves are reported, so that only one of [Formula: see text] or [Formula: see text] is physically measured, and particles appear for [Formula: see text] and de Broglie waves occur for [Formula: see text], but in either event both a measurable and an immeasurable energy exists. Conventional quantum mechanics operates under circumstances such that [Formula: see text] vanishes and [Formula: see text] becomes purely imaginary. If both [Formula: see text] and [Formula: see text] are generated as the gradient of a potential, the total particle energy is necessarily conserved in the conventional manner.

Modeling of macroscopic quantum states in functional properties of the laser-induced 4D-topological nanoclusters in thin films on solid surface

Nanocluster structures can be easily modified in necessary direction and by controlled way in femtonanophotonics experiments. The variation of the key topology parameters can result in new type of the quantum correlation states/size effect for charged particles. In our earlier experiments we studied laser-induced topological nanoclusters structures of different types in thin films with unique phenomena in electrophysics and optics (see [1-3]). A simple 2-steps mechanism for enhancement of quantum behavior (e.g. in electroconductivity) exists for different conditions. First, when inelastic length linelastic > acluster we have no incoherent electron-phonon (e-ph) scattering, i.e. the coherent process takes place. Second, when de Broglie wave length λdB ≡ ℓcoh < Λ, (acluster – cluster size , Λ – spatial period of nanoparticle distribution) the coherent tunneling without loss occurs, and a long-range order with interference of the states takes place in the medium due to lattice structure.

De Broglie Wave Analysis of the Heisenberg Uncertainty Minimum Limit under the Lorentz Transformation

A simple analysis using differential calculus has been done to consider the minimum limit of the Heisenberg uncertainty principle in the relativistic domain. An analysis is made by expressing the form of and based on the Lorentz transformation, and their corresponding relation according to the de Broglie wave packet modification. The result shows that in the relativistic domain, the minimum limit of the Heisenberg uncertainty is p x ?/2 and/or E t ?/2, with is the Lorentz factor which depend on the average/group velocity of relativistic de Broglie wave packet. While, the minimum limit according to p x ?/2 or E t ?/2, is the special case, which is consistent with Galilean transformation. The existence of the correction factor signifies the difference in the minimum limit of the Heisenberg uncertainty between relativistic and non-relativistic quantum. It is also shown in this work that the Heisenberg uncertainty principle is not invariant under the Lorentz transformation. The form p x ?/2 and/or E t ?/2 are properly obeyed by the Klein-Gordon and the Dirac solution. Key words: De Broglie wave packet, Heisenberg uncertainty, Lorentz transformation, and minimum limit.