Ciro Santilli often wonders to himself, how much of the natural sciences can one learn in a lifetime? Certainly, a very strong basis, with concrete experimental and physics, chemistry and biology should be attainable to all? How much Ciro manages to learning and teach in those areas is a kind of success metric of Ciro's life.

Physics is the art of predicting the future by modelling the world with mathematics.

Ciro Santilli doesn't know physics. He writes about it partly to start playing with some scientific content for: OurBigBook.com, partly because this stuff is just amazingly beautiful.

Ciro's main intellectual physics fetishes are to learn quantum electrodynamics (understanding the point of Lie groups being a subpart of that) and condensed matter physics.

Every science is Physics in disguise, but the number of objects in the real world is so large that we can't solve the real equations in practice.

Luckily, due to emergence, we can use uglier higher level approximations of the world to solve many problems, with the complex limits of applicability of those approximations.

As of 2019, all known physics can be described by two theories:

Unifying those two into the theory of everything one of the major goals of modern physics.

The approach many courses take to physics, specially "modern Physics" is really bad, this is how it should be taught:

- start by describing experiments that the previous best theory did not explain, see also: Section "Physics education needs more focus on understanding experiments and their history"
- then, give the final formula for the next best theory
- then, give all the important final implications of that formula, and how it amazingly describes the experiments. In particular this means: doing physics means calculating a number
- then, give some mathematical intuition on the formulas, and how the main equation could have been derived
- finally, then and only then, start deriving the outcomes of the main formula in detail

This is likely because at some point, experiments get more and more complicated, and so people are tempted to say "this is the truth" instead of "this is why we think this is the truth", which is much harder.

But we can't be lazy, there is no replacement to the why.

Related:

- settheory.net/learnphysics and www.youtube.com/watch?v=5MKjPYuD60I&list=PLJcTRymdlUQPwx8qU4ln83huPx-6Y3XxH from settheory.net
- math.ucr.edu/home/baez/books.html by John Baez. Mentions:
This webpage doesn't have lots of links to websites. Websites just don't have the sort of in-depth material you need to learn technical subjects like advanced math and physics — at least, not yet. To learn this stuff, you need to read lots of books

Ciro Santilli is trying to change that: OurBigBook.com. - web.archive.org/web/20210324182549/http://jakobschwichtenberg.com/one-thing/ by Jakob Schwichtenberg

This is the only way to truly understand and appreciate the subject.

Understanding the experiments gets intimately entangled with basically learning the history of physics, which is extremely beneficial as also highlighted by Ron Maimon, related: there is value in tutorials written by early pioneers of the field.

"How we know" is a basically more fundamental point than "what we know" in the natural sciences.

In the Surely You're Joking, Mr. Feynman chapter O Americano, Outra Vez! Richard Feynman describes his experience teaching in Brazil in the early 1950s, and how everything was memorized, without any explanation of the experiments or that the theory has some relationship to the real world!

Although things have improved considerably since in Brazil, Ciro still feels that some areas of physics are still taught without enough experiments described upfront. Notably, ironically, quantum field theory, which is where Feynman himself worked.

Feynman gave huge importance to understanding and explaining experiments, as can also be seen on Richard Feynman Quantum Electrodynamics Lecture at University of Auckland (1979).

Everyone is beginner when the field is new, and there is value in tutorials written by beginners.

For example, Ciro Santilli felt it shocking how direct and satisfying Richard Feynman's scientific vulgarization of quantum electrodynamics were, e.g. at: Richard Feynman Quantum Electrodynamics Lecture at University of Auckland (1979), and that if he had just assumed minimal knowledge of mathematics, he was about to give a full satisfactory picture in just a few hours.

The same also applies to early original papers of the field, as notably put forward by Ron Maimon.

In Physics, in order to test a theory, you must be able to extract a number from it.

It does not matter how, if it is exact, or numerical, or a message from God: a number has to come out of the formulas in the end, and you have to compare it with the experimental data.

Many theoretical physicists seem to forget this in their lectures, see also: Section "How to teach and learn physics".

Nature is a black box, right?

You don't need to understand the from first principles derivation of every single phenomena.

And most important of all: you should not start learning phenomena by reading the from first principles derivation.

Instead, you should see what happens in experiments, and how matches some known formula (which hopefully has been derived from first principles).

Only open the boxes (understand from first principles derivation) if the need is felt!

E.g.:

- you don't need to understand everything about why SQUID devices have their specific I-V curve curve. You have to first of all learn what the I-V curve would be in an experiment!
- you don't need to understand the fine details of how cavity magnetrons work. What you need to understand first is what kind of microwave you get from what kind of input (DC current), and how that compares to other sources of microwaves
- lasers: same

Physics is all about predicting the future. If you can predict the future with an end result, that's already predicting the future, and valid.

Videos should be found/made for all of those: videos of all key physics experiments

- speed of light experiment
- basically all experiments listed under Section "Quantum mechanics experiment" such as:
- Davisson-Germer experiment

This shows that viewing electromagnetism as gauge theory does have experimentally observable consequences. TODO understand what that means.

In more understandable terms, it shows that the magnetic vector potential matters where the magnetic field is 0.

Classic theory predicts that the output frequency must be the same as the input one since the electromagnetic wave makes the electron vibrate with same frequency as itself, which then irradiates further waves.

But the output waves are longer because photons are discrete and energy is proportional to frequency:

The formula is exactly that of two relativistic billiard balls colliding.

Therefore this is evidence that photons exist and have momentum.

No matter how hight the wave intensity, if it the frequency is small, no photons are removed from the material.

This is different from classic waves where energy is proportional to intensity, and coherent with the existence of photons and the Planck-Einstein relation.

The key is to define only the minimum number of measures: if you define more definitions become over constrained and could be inconsistent.

Learning the modern SI is also a great way to learn some interesting Physics.

Great overview of the earlier history of unit standardization.

Gives particular emphasis to the invention of gauge blocks.

TODO how does basing it on the elementary charge help at all? Can we count individual electrons going through a wire? www.nist.gov/si-redefinition/ampere/ampere-quantum-metrology-triangle by the NIST explains that is it basically due to the following two quantized solid-state physics phenomena/experiments that allows for extremely precise measurements of the elementary charge:

- quantum Hall effect, which has discrete resistances of type:
for integer values of $ν$.$R_{xy}=I_{channel}V_{Hall} =e_{2}νh $
- Josephson effect, which provides the Josephson constant which equals:
$K_{J}=h2e $

Unit of electric current.

Affected by the ampere in the 2019 redefinition of the SI base units.

Unit of mass.

Defined in the 2019 redefinition of the SI base units via the Planck constant. This was possible due to the development of the kibble balance.

Measures weight from a voltage.

Named after radio pioneer Heinrich Hertz.

Uses the frequency of the hyperfine structure of caesium-133 ground state, i.e spin up vs spin down of its valence electron $6s_{1}$, to define the second.

International System of Units definition of the second since 1967, because this is what atomic clocks use.

TODO why does this have more energy than the hyperfine split of the hydrogen line given that it is further from the nucleus?

Highlighted at the Origins of Precision by Machine Thinking (2017).

A series of systems usually derived from the International System of Units that are more convenient for certain applications.

Currently an informal name for the Standard Model

Chronological outline of the key theories:

- Maxwell's equations
- Schrödinger equation
- Date: 1926
- Numerical predictions:
- hydrogen spectral line, excluding finer structure such as 2p up and down split: en.wikipedia.org/wiki/Fine-structure_constant

- Dirac equation
- Date: 1928
- Numerical predictions:
- hydrogen spectral line including 2p split, but excluding even finer structure such as Lamb shift

- Qualitative predictions:
- Antimatter
- Spin as part of the equation

- quantum electrodynamics
- Date: 1947 onwards
- Numerical predictions:
- Qualitative predictions:
- Antimatter
- spin as part of the equation

As of the 20th century, this can be described well as "the phenomena described by Maxwell's equations".

Back through its history however, that was not at all clear. This highlights how big of an achievement Maxwell's equations are.

Unified all previous electro-magnetism theories into one equation.

Explains the propagation of light as a wave, and matches the previously known relationship between the speed of light and electromagnetic constants.

The equations are a limit case of the more complete quantum electrodynamics, and unlike that more general theory account for the quantization of photon.

The equations are a system of partial differential equation.

The system consists of 6 unknown functions that map 4 variables: time t and the x, y and z positions in space, to a real number:and two known input functions:

- $E_{x}(t,x,y,z)$, $E_{y}(t,x,y,z)$, $E_{z}(t,x,y,z)$: directions of the electric field $E:R_{4}→R_{3}$
- $B_{x}(t,x,y,z)$, $B_{y}(t,x,y,z)$, $B_{z}(t,x,y,z)$: directions of the magnetic field $B:R_{4}→R_{3}$

- $ρ:R_{3}toR$: density of charges in space
- $J:R_{3}→R_{3}$: current vector in space. This represents the strength of moving charges in space.

Due to the conservation of charge however, those input functions have the following restriction:

$∂t∂ρ +∇⋅J=0$

Also consider the following cases:

- if a spherical charge is moving, then this of course means that $ρ$ is changing with time, and at the same time that a current exists
- in an ideal infinite cylindrical wire however, we can have constant $ρ$ in the wire, but there can still be a current because those charges are movingSuch infinite cylindrical wire is of course an ideal case, but one which is a good approximation to the huge number of electrons that travel in a actual wire.

The goal of finding $E$ and $B$ is that those fields allow us to determine the force that gets applied to a charge via the Equation "Lorentz force", and then to find the force we just need to integrate over the entire body.

Finally, now that we have defined all terms involved in the Maxwell equations, let's see the equations:

$divE=ε_{0}ρ $

$divB=0$

$∇×E=−∂t∂B $

$∇×B=μ_{0}(J+ε_{0}∂t∂E )$

You should also review the intuitive interpretation of divergence and curl.

$force_density=ρE+J×B$

A little suspicious that it bears the name of Lorentz, who is famous for special relativity, isn't it? See: Maxwell's equations require special relativity.

For numerical algorithms and to get a more low level understanding of the equations, we can expand all terms to the simpler and more explicit form:

$∂x∂E_{x} +∂x∂E_{y} +∂x∂E_{z} =ε_{0}ρ ∂x∂B_{x} +∂x∂B_{y} +∂x∂B_{z} =0∂y∂E_{z} −∂z∂E_{y} =−∂t∂B_{x} ∂z∂E_{x} −∂x∂E_{z} =−∂t∂B_{y} ∂x∂E_{y} −∂y∂E_{x} =−∂t∂B_{z} ∂y∂B_{z} −∂z∂B_{y} =μ_{0}(J_{x}+ε_{0}∂t∂E_{x} )∂z∂B_{x} −∂x∂B_{z} =μ_{0}(J_{y}+ε_{0}∂t∂E_{y} )∂x∂B_{y} −∂y∂B_{x} =μ_{0}(J_{z}+ε_{0}∂t∂E_{z} )$

As seen from explicit scalar form of the Maxwell's equations, this expands to 8 equations, so the question arises if the system is over-determined because it only has 6 functions to be determined.

As explained on the Wikipedia page however, this is not the case, because if the first two equations hold for the initial condition, then the othe six equations imply that they also hold for all time, so they can be essentially omitted.

It is also worth noting that the first two equations don't involve time derivatives. Therefore, they can be seen as spacial constraints.

TODO: the electric field and magnetic field can be expressed in terms of the electric potential and magnetic vector potential. So then we only need 4 variables?

Static case of Maxwell's law for electricity only.

The "static" part is important: if this law were true for moving charges, we would be able to transmit information instantly at infinite distances. This is basically where the idea of field comes in.

In the standard formulation of Maxwell's equations, the electric current is a convient but magic input.

Would it be possible to use Maxwell's equations to solve a system of pointlike particles such as electrons instead?

The following suggest no, or only for certain subcases less general than Maxwell's equations:

This is the type of thing where the probability aspect of quantum mechanics seems it could "help".

TODO it would be awesome if we could de-generalize the equations in 2D and do a JavaScript demo of it!

Not sure it is possible though because the curl appears in the equations:

TODO: I'm surprised that the Wiki page barely talks about it, and there are few Google hits too! A sample one: www.researchgate.net/publication/228928756_On_the_existence_and_uniqueness_of_Maxwell's_equations_in_bounded_domains_with_application_to_magnetotellurics

In the context of Maxwell's equations, it is vector field that is one of the inputs of the equation.

Section "Maxwell's equations with pointlike particles" asks if the theory would work for pointlike particles in order to predict the evolution of this field as part of the equations themselves rather than as an external element.

Measured in amperes in the International System of Units.

After the 2019 redefinition of the SI base units it is by definition exactly $1.60217663410_{−19}$ Joules.

The voltage changes perpendicular to the current when magnetic field is applied, Just watch this:

Applications:

- the direction of the effect proves that electric currents in common electrical conductors are made up of negative charged particles
- measure magnetic fields, TODO vs other methods

Other more precise non-classical versions:

A different and more elegant way to express Maxwell's equations by using the:instead of the:

There are several choices of electromagnetic four-potential that lead to the same physics.

E.g. thinking about the electric potential alone, you could set the zero anywhere, and everything would remain be the same.

The Lorentz gauge is just one such choice. It is however a very popular one, because it is also manifestly Lorentz invariant.

Alternative to the Lorentz gauge, but less used in general as it is not as nice for relativity invariance.

Implementations:

- Hall effect based
- SQUID device

Explains how it is possible that everyone observes the same speed of light, even if they are moving towards or opposite to the light!!!

This was first best observed by the Michelson-Morley experiment, which uses the movement of the Earth at different times of the year to try and detect differences in the speed of light.

This leads leads to the following conclusions:

- to length contraction and time dilation
- the speed of light is the maximum speed anything can reach

All of this goes of course completely against our daily Physics intuition.

The "special" in the name refers to the fact that it is a superset of general relativity, which also explains gravity in a single framework.

Since time and space get all messed up together, you have to be very careful to understand what it means to say "I observed this to happen over there at that time", otherwise you will go crazy. A good way to think about is this:

- use Einstein synchronization to setup a bunch of clocks for every position in your frame of reference
- on every point of space, you put a little detector which records events and the time of the event
- each detector can only detect events locally, i.e. events that happen very close to the detector
- then, after the event, the detectors can send a signal to you, who is sitting at the origin, telling you what they detected

This single experimental observation/idea is the basis for all of special relativity.

Special relativity is the direct result of people bending their backs to accommodate for this really weird fact.

Bibliography:

- Subtle is the Lord by Abraham Pais (1982) chapter III "Relativity, the special theory" has a good sketch as you may imagine.

Can you just imagine what if luminiferous aether was one single fixed rigid body? This is apparently what Maxwell believed, Subtle is the Lord by Abraham Pais (1982) page 111 quoting his entry to Encyclopedia Britannica:

There can be no doubt that the interplanetary and interstellar spaces are not empty but are occupied by a material substance or body, which is certainly the largest, and probably the most uniform, body of which we have any knowledge.Then it would provide a natural space coordinate for the entire universe!

Apparently Einstein was the first to completely say: let's just screw this aether thing completely then, it's getting too complicated, and we don't really need it. As Wikipedia puts it well, in very unencyclopedic tone

^{[ref]}: Aether fell to Occam's razor.Given experiments such as the Fizeau experiment and the Michelson-Morley experiment that couldn't really detect the Earth's movement across aether, people started to wonder if the Earth wasn't dragging the luminiferous aether.

- moving magnet and conductor problem: the more experiments confirm Maxwell's equations, the more special relativity has to be correct
- aberration TODO more precisely how it is evidence.

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