If you have a PDE that models physical phenomena, it is fundamental that:

Unlike for ordinary differential equations which have the Picard–Lindelöf theorem, the existence and uniqueness of solution is not well solved for PDEs.

For example, Navier-Stokes existence and smoothness was one of the Millennium Prize Problems.

Often just called collimated light due to the collimator being the main procedure to obtain it.

However, you move very far away from the source, e.g. the Sun, you also get essentially parallel light.

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Given the view of the Standard Model where the electron and quarks are just completely separate matter fields, there is at first sight no clear theoretical requirement for that.

As mentioned e.g. at QED and the men who made it: Dyson, Feynman, Schwinger, and Tomonaga by Silvan Schweber (1994) chapter 1.6 "Hole theory", Dirac initially wanted to think of the holes in his hole theory as the protons, as a way to not have to postulate a new particle, the positron, and as a way to "explain" the proton in similar terms. Others however soon proposed arguments why the positron would need to have the same mass, and this idea had to be discarded.

The Dirac equation can be derived basically "directly" from the Representation theory of the Lorentz group for the spin half representation, this is shown for example at Physics from Symmetry by Jakob Schwichtenberg (2015) 6.3 "Dirac Equation".

The Diract equation is the spacetime symmetry part of the quantum electrodynamics Lagrangian, i.e. is describes how spin half particles behave without interactions. The full quantum electrodynamics Lagrangian can then be reached by adding the $U(1)$ internal symmetry.

As mentioned at spin comes naturally when adding relativity to quantum mechanics, this same method allows us to analogously derive the equations for other spin numbers.

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These videos can give some geometric insight and do have their value.

But they are sometimes too slow, there are never any mention of experiments, just "the truth".

And when things get "mathy", it sticks to a more qualitative view which may not be enough.

Very over the top with sexy demons and angels making appearances, and has some classic aesthetic artistic value :-)

The inaugural that predicted the Josephson effect.

Published on Physics Letters, then a new journal, before they split into Physics Letters A and Physics Letters B. True Genius: The Life and Science of John Bardeen mentions that this choice was made rather than the more prestigious Physical Review Letters because they were not yet so confident about the results.

Paywalled by Elsevier as of 2023 at: www.sciencedirect.com/science/article/abs/pii/0031916362913690

A tiny idealized magnet! It is a very good model if you have a small strong magnet interacting with objects that are far away, notably other magnetic dipoles or a constant magnetic field.

The cool thing about this model is that we have simple explicit formulas for the magnetic field it produces, and for how this little magnet is affected by a magnetic field or by another magnetic dipole.

This is the perfect model for electron spin, but it can also be representative of macroscopic systems in the right circumstances.

The intuition for the name is likely that "dipole" means "both poles are on the same spot".

2023 Silk Road's Second-in-Command Gets 20 Years in Prison www.wired.com/story/silk-road-variety-jones-sentencing/

2016 "Exclusive: Our Thai prison interview with the alleged top advisor to Silk Road" arstechnica.com/tech-policy/2016/09/exclusive-our-thai-prison-interview-with-an-alleged-top-advisor-to-silk-road/

2015 The Variety Show On the trail of the man believed to be Variety Jones, one of the architects of the defunct drug marketplace Silk Road. www.vice.com/en/article/wnx5qn/the-variety-show

www.justice.gov/usao-sdny/file/797251/download some kind of case file of his trial.

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The curious thing about VJ is that he actually has some culture and says cool things, e.g.:

Chopsticks is a hand game typically played by two or more players. It's a game that involves using fingers to represent numbers, and it can be played with both strategy and skill. The objective is to eliminate all of your opponents' "fingers" (or hands) by touching them and using simple rules of movement and counting. ### Basic Rules: 1. **Starting Position**: Each player starts with one finger extended on each hand (usually two hands).

"Kaye and Laby" refers to the "Kaye and Laby: Tables of Physical and Chemical Constants," which is a reference book widely used in the fields of physics and chemistry. The book contains a comprehensive collection of tables that provide various physical and chemical constants, properties of materials, and other essential data that researchers and scientists often require. Originally compiled by Sir J. H. Kaye and Sir D. W.