Andrew, Jones etal,-- The magnetic Aharonov–Bohm effect is also closely related to Dirac's<https://en.wikipedia.org/wiki/Paul_Dirac> argument that the existence of a magnetic monopole<https://en.wikipedia.org/wiki/Magnetic_monopole> can be accommodated by the existing magnetic source-free Maxwell's equations<https://en.wikipedia.org/wiki/Maxwell%27s_equations> if both electric and magnetic charges are quantized.
A magnetic monopole implies a mathematical singularity in the vector potential, which can be expressed as a Dirac string<https://en.wikipedia.org/wiki/Dirac_string> of infinitesimal diameter that contains the equivalent of all of the 4πg flux from a monopole "charge" g. The Dirac string starts from, and terminates on, a magnetic monopole. Thus, assuming the absence of an infinite-range scattering effect by this arbitrary choice of singularity, the requirement of single-valued wave functions (as above) necessitates charge-quantization. That is, 2 q e q m ℏ c {\displaystyle 2{\frac {q_{\text{e}}q_{\text{m}}}{\hbar c}}} [2{\frac {q_{{\text{e}}}q_{{\text{m}}}}{\hbar c}}] must be an integer (in cgs<https://en.wikipedia.org/wiki/Gaussian_units> units) for any electric charge qe and magnetic charge qm. Like the electromagnetic potential<https://en.wikipedia.org/wiki/Electromagnetic_potential> A the Dirac string is not gauge invariant (it moves around with fixed endpoints under a gauge transformation) and so is also not directly measurable. Bob Cook Sent from Mail<https://go.microsoft.com/fwlink/?LinkId=550986> for Windows 10 ________________________________ From: JonesBeene <jone...@pacbell.net> Sent: Wednesday, August 7, 2019 5:17:54 AM To: vortex-l@eskimo.com <vortex-l@eskimo.com> Subject: RE: [Vo]:FW: coherent system energy states Andrew, Bob A good paper on this subject (longitudinal waves) is “Unravelling the potentials puzzle and corresponding case for the scalar longitudinal electrodynamic wave” Donald Reed 2019 J. Phys.: Conf. Ser. 1251 012043 Reed does not make the scalar to neutrino connection, which seems to serve the same purposes, which is to explore the line between what is real and what seems real because it balances equations. The best thing one can say about QM is that it lends physical credulity to an imaginary world… but then again, what is real? From: Andrew Meulenberg<mailto:mules...@gmail.com> …. At the short distance of deep-orbits from the nucleus, the neutrino (considered to be similar to photons) would be in the "longitudinal photon" mode
