This form is not really an inner product in the common modern definition, because it is not positive definite, only a symmetric bilinear form.

The history of light if funny.

First people thought it was a particle, as per corpuscular theory of light, notably Newton supported the corpuscular theory of light.

But then evidence of the diffraction of light start to become unbearably strong, culminating in the Arago spot.

And finally it was understood from Maxwell's equations that light is a form of electromagnetic radiation, as its speed was perfectly predicted by the theory.

But then evidence of particle nature started to surface once again with the photoelectric effect. Physicists must have been driven mad by all these changes.

Bibliography:

Can be approximated with a diaphragm.

Funding:

This section discusses the pre-photon understanding of the polarization of light. For the photon one see: photon polarization.

polarization.com/history/history.html is a good page.

People were a bit confused when experiments started to show that light might be polarized. How could a wave that propages through a 3D homgenous material like luminiferous aether have polarization?? Light would presumably be understood to be analogous to a sound wave in 3D medium, which cannot have polarization. This was before Maxwell's equations, in the early 19th century, so there was no way to know.

The growing number of parameters of the Standard Model is one big source of worry for early 21st century physics, much like the growing number of particles was a worry in the beginning of the 20th (but that one was solved by 2020).

List: en.wikipedia.org/w/index.php?title=Standard_Model&oldid=1042518939#Construction_of_the_Standard_Model_Lagrangian

TODO original experiment?

Laughlin paper: 1981 Quantized Hall conductivity in two dimensions.

Shows a cool new type of matter: Abelian anyons.

so the solution is:We notice that the solution has continuous spectrum, since any value of $E$ can provide a solution.

Contains the full state of the quantum system.

This is in contrast to classical mechanics where e.g. the state of mechanical system is given by two real functions: position and speed.

The wave equation in position representation on the other hand encodes speed in "how fast does the complex phase spin around", and direction in "does it spin clockwise or counterclockwise", as described well at: Video "Visualization of Quantum Physics (Quantum Mechanics) by udiprod (2017)". Then once you understand that, it is more compact to just view those graphs with the phase color coded as in Video "Simulation of the time-dependent Schrodinger equation (JavaScript Animation) by Coding Physics (2019)".

The first really good quantum mechanics theory made compatible with special relativity was the Dirac equation.

And then came quantum electrodynamics to improve it: Dirac equation vs quantum electrodynamics.

TODO: does it use full blown QED, or just something intermediate?

www.youtube.com/watch?v=NtnsHtYYKf0 "Mercury and Relativity - Periodic Table of Videos" by Periodic Videos (2013). Doesn't give the key juicy details/intuition. Also mentioned on Wikipedia: en.wikipedia.org/wiki/Relativistic_quantum_chemistry#Mercury

This one might actually be understandable! It is what Richard Feynman starts to explain at: Richard Feynman Quantum Electrodynamics Lecture at University of Auckland (1979).

The difficulty is then proving that the total probability remains at 1, and maybe causality is hard too.

The path integral formulation can be seen as a generalization of the double-slit experiment to infinitely many slits.

Feynman first stared working it out for non-relativistic quantum mechanics, with the relativistic goal in mind, and only later on he attained the relativistic goal.

TODO why intuitively did he take that approach? Likely is makes it easier to add special relativity.

This approach more directly suggests the idea that quantum particles take all possible paths.

Bibliography review:

Course outline given:

Non-relativistic QFT is a limit of relativistic QFT, and can be used to describe for example condensed matter physics systems at very low temperature. But it is still very hard to make accurate measurements even in those experiments.

Defines "relativistic" as: "the Lagrangian is symmetric under the Poincaré group".

Mentions that "QFT is hard" because (a finite list follows???):But I guess that if you fully understand what that means precisely, QTF won't be too hard for you!

Notably, this is stark contrast with rotation symmetry groups (SO(3)) which appears in space rotations present in non-relativistic quantum mechanics.

www.youtube.com/watch?v=T58H6ofIOpE&t=5097 describes the relativistic particle in a box thought experiment with shrinking walls

Dirac field.