However, creation and annihilation of a massless particle, like the photon, is not so simple because c is the constant velocity for both group and phase velocities. By ignoring diffraction, the transverse spatial coherence then becomes c multiplied by the temporal coherence. This photon is described in Maxwell’s theory by two real, sinusoidal, wave functions on mutually normal planes, one lagging in phase by π/2. Reduction of the packet thus occurs within its spatio-temporal coherence. From Equation (2), the longitudinal coherence length is found to be 2cσ; the coherence time 2σ/c. Through σ, the coherence values depend on initial conditions for the transmitting wave packet.

What can be understood about the transverse spatial coherence? Can a wave front become wider than the coherence? Does the coherence limit beam spread? Compare the temporal coherence from section 2.1 (2σ/c) with the transverse spatial coherence from Section 2.2 (tan^{−1}(λ/d)). It is clear that when, typically
$\lambda \ll d$, the ratio
$c\Delta t/\left(\lambda /d\right)~2\sigma d/\lambda $. Thus, when
$\sigma >d/\lambda $, the transverse spatial coherence is determined by longitudinal temporal coherence; otherwise wave diffraction, d/λ, determines the transverse coherence. These features are in principle demonstrated by comparing diffraction of light generated by a laser source or point source. Supercurrents are massive, so we can now consider their reduction in their rest frame, i.e. in Newtonian time.

5. Super Currents

The solution for superconductivity was given by Bardeen, Cooper and Schrieffer [19] [20] [21]. Electrons in metallic superconductors occupy two states, normal and superconducting, i.e. at temperatures below critical T < T_{c}, and applied magnetic field H < H_{c}. The superconducting state consists of a pair of electrons bound together by a lattice distortion and energy gap ΔE. This Cooper pair has zero net momentum K = 0; zero net spin S = 0^{5}; and electronic charge 2e. The wave is superconductive, with zero resistivity ρ = 0 and therefore zero internal electric field E = 0. Meanwhile zero internal magnetic field induction B = 0 is demonstrated by the Meissner-Ochsenfeld effect. This is the basic state, though minor complexity arises in type II superconductors where electrons propagate quantized magnetic flux lines, and also at normal state boundaries as in the Josephson dc and ac tunneling effects. Since the normal and superconducting states are, by supposition, time independent, anomalies naturally arise in their dynamics. In high temperature superconductors, we understand the Cooper pair to be Wannier excitons with charge Q = 0 [1]. This is an added obstacle for conductivity. It suggests a real wave function that is Bosonic and commensurate with the lattice. How does transmission of current occur?

Consider for contrast, the more comprehensible oscillator strength in spectral transitions. This is a calculation of the relative transition rate between two stable electronic states, initial and final, excited by an operator O:
${\left|\langle {y}_{f}|O|{y}_{i}\rangle \right|}^{2}$, typically integrated over space, time, and density of states. In a dipole transition for example, O = er, where e represents electronic charge. Apply such a transition to measurement of resistivity of a superconductor by the four probe technique: at T < T_{c} and H < H_{c}. When a voltage is applied between two outer terminals the supercurrent is measured at the inner sensor terminals. This current consists of normal electric charges that emerge from the positive terminal of a superconductor containing no electromagnetic fields. In dispersion dynamics this feature is not surprising: the terminals supply energy that breaks the Cooper pairs which release their normal charges to metallic terminals in Newtonian time, i.e. when considered in the rest frame. The dynamics will be governed by the transfer of thermal or electrical energy to the Cooper pairs. Each pair has zero net momentum, while any dynamic momentum that is transferred to the normal electrons at terminals, can be obtained by the pairs from the massive lattice as in X-ray diffraction and in the Mossbauer effect. Between the terminals, a supercurrent flows in absence from any internal electric or magnetic fields.

6. Summary and Conclusion

In dispersion dynamics, the product of the group velocity and phase velocity in a wave packet is equal to the square of the speed of light. In consequence, uncertainty is not an arbitrary limit, but a calculated expectation that has varied effects in reduction. Moreover, since immobile positive ions are not deflected by the magnetic Lorentz force, the positive Hall coefficients that are measured in certain metals and doped semiconductors must be generated by (negatively charged) “holey” electrons that exist in states having peculiar dispersive dynamics. Furthermore, in ionic high temperature superconductors, those electrons are supposedly contained in bosonic pairs of Wannier excitons. How then do supercurrents flow in the absence of internal electromagnetic fields and with zero momentum, zero spin, and zero net charge? In dispersion dynamics, the flow occurs by the reduction of time-independent waves that are consistent with packet decay, i.e. in Newtonian time between terminals. This reduction is a new solution for an unanswered problem. Following the analysis, the reduction is in principle verifiable by time-dependent measurements. Superconductivity is a physical paradigm for Boson statistics, for condensation, and for related phenomena that occur throughout the broad scope of physic. These further applications of dispersion dynamics will be objects for further study.

NOTES

^{1}The group and phase velocities supposedly combine to produce Dirac’s “jitter” [Dirac, P.A.M., The Principles of Quantum Mechanics (1958) 4th Edition, Oxford] without purportedly offending the requirement for infinite energy in his electron particle, that he claims to have velocity c.

^{2}Dirac supposed point particles apparently because the wave packet was, to him, unstable [ibid.].

^{3}Pais [4] wrote that “It was [Einstein’s] almost solitary conviction that quantum mechanics is logically consistent, but that it is an incomplete manifestation of an underlying theory in which an underlying objective reality is possible.” Actually, in his EPR paper [Einstein, A., Podolsky, B. and Rosen, N., Phys.Rev. 47 777-780 (19350] Einstein held that Bohr’s theory is incomplete. The latter had claimed that all that can be known about an electron is in its wave function, particularly with regard to momentum and position. This seems an unlikely theory since Gödel’s mathematical theorems on completeness and consistency in axiomatic systems [Gödel, K., Monatshefte für Mathematik und Physik 38 173-198 doi: 10.1007/BF01700692]. Popper was another realist in common with Einstein [Popper, K.R., Quantum theory and the schism in physics, Ed.Bartley W.W. III, (1982) Hutchinson]. This footnote does not contribute to the debate so much as outline background for arguments given in the text.

^{4}This description has the same Hall result as previously [1] [7], but equations 10 and 11 had been misapplied to the magnetic field instead of the electric field.

^{5}Even the intrinsic spin is problematic in dynamics. Spin is only measured in the presence of a magnetic field, so it is not obvious whether a measured magnetic moment is induced or “intrinsic” [1] [9]. Nor is it obvious that the electron is a point particle: it has magnetic moment with dimensions that include area? In particular, how can an “intrinsic spin” exist in the absence of a circulating current? Moreover, what physical transition is involved when a spin flip occurs? In dispersion dynamics, the spin is induced in non-resistive phase currents.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

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